Boy, this is one of my favorite topics and if you feel so inclined (no pun intended) you can read a short little blog post about it here. 

There are a few things you can do to improve your hill-climbing experience.

1. Avoid hills! Sometimes this may mean going out of your way a little bit, but if you know you have a hilly commute or ride in your future, plan ahead and leave the house a bit early to circumvent the evil grade. 

2. If an unavoidable hill in your future, approaching it the right way will help immensely. Before you get to the bottom of the hill, think about your speed and how you will best carry your momentum onto the incline. Coming into a climb at a good pace will help you climb more efficiently and will get you up and over the top faster. A wise person once said "Momentum is the sharpest arrow in your quiver, maintain it and it will serve you well". 

3. Figure out what mindset best puts you in a positive hill-climbing mood. For some this means finding your "zen" and existing outside of your body and outside of the pain in your legs. For others this may be cursing the hill the whole way up while scowling and spitting occasionally. For myself, I find that each hill usually requires a different mindset, since each climb is it's own unique experience. 

4. Pick a spot and get to it! This trick has helped me over many a hill. All you do is pick a spot up ahead of you on the hill, a landmark of some sort (a street sign, a mailbox, a pretty flower, another cyclist, or perhaps a pub), and get to it. Once you get to it, pick another landmark further up ahead and repeat the process (unless of course, the first landmark that you picked was a cafe or bar, then you should probably stop and indulge yourself). 

5. Figure out what cadence works best for you. Do you enjoy spinning your legs faster, in an easier gear? Or do you prefer to mash out a harder gear at a slower RPM. Most people have a specific cadence range that best suits their body/cardiovascular system/fitness level/etc. Or, if you are the type that rides with headphones in, match your cadence to the beat of your music and then pretend that you are in a music video. That one always works.

6. Remind yourself that you will eventually get up and over the hill and that life will move on.

Number one advice, which I see people not following every day in my hilly neighborhood: USE YOUR GEARS. That's what they're there for. Every day I see people with 21 or 24 speeds on their bike who never shift out of the one they're always in.

Ideally you should be maintaining the same high cadence (crank RPM) all the time, whether traveling up, down, or on the flat -- 60 to 80 RPM. Shift down before you get there, to your lowest gear, and hold that cadence up as long as you can. On the worst hills you can't, but you'll struggle less in your lowest gear than you will in some middle grinding position.

If you want, you can go in and have a super-small chain ring put on the front (or a super-big one on the innermost spot on the back). A decent bike shop can get you set up so it's an easier pedal but still works with your derailleur.

Beer legs :)

No, but seriously though. Hills are my arch nemesis. I used to have to ride up Stone every single day, and I never looked forward to it. It always hurt, I was always pissed off doing it, and it usually kicked up my asthma pretty bad.

I got into the habit of taking side roads and zig-zagging a little bit. It took a bit longer, but I also found that I wasn't pushing myself so hard or riding so aggressively when I didn't have an "audience" of other cyclists and cars.

I also always tell myself that I can get off and walk if I need to. I have never, not even once actually done that- but allowing myself that option makes me feel a little less aggravated about it. I do sometimes stop to drink some water and catch my breath if I need to- and I've never had to do that more than once on any hill.

Another method, if you're feeling competitive- is to find someone in front of you who is riding slower than you are. Work to pass them- and then work to make sure they don't pass you again. The little dose of competition is often enough to get me to step it up a notch and more or less enjoy the pain.

Also, I highly recommend either clipless pedals or power grips if you're not already. Being able to pull up, and use a different muscle group really helped me get over hills.

What helped me the most was to surrender myself to the hills and embrace them as wonderful. Hills are beautiful, and I am lucky to have opportunity to climb them on my bicycle. Sure, they make me slower and make me tired, they exacerbate my hunger and make me sweaty, but they are massive and powerful. You are moving slower, so take the opportunity to look at your surroundings more closely than you would be able to going the opposite direction. Look at how the trees are rejecting the pavement, roots pushing up and making cracks. Look at how the view changes as you gain altitude. You are drastically changing your place in relation to the water as you climb. Think about how powerful that experience is, and you might not feel like you are grinding away against nothing anymore.

Consider an electric bike.

I live at the top of a steep hill, and that's what made a big difference for me. I've now ridden a couple thousand miles mostly commuting over the last year, and saved a lot of CO2 producing combustion as a result.

I am still legal for using the often superior right of way a bicycle provides, such as crossing I-90 when Mariners fans are all trying to drive to Safeco Field.

I am surprised that electric bikes haven't caught on more considering how well they can work, particularly with the newer batteries now out.

Google Bicycle Directions takes terrain into account, and will route you around hills and avoid steep slopes.  It can be a big help in finding parallel routes that have gentler grades.  I always check it now before going somewhere new.

I've heard others say toeclips or pedal straps help.  If you can push in a full circle with both legs, rather than just down with one leg at a time, you can use more muscles.  But I don't use any of that, I just go slow.  

I shift way down and keep a steady 80 rpm or so. Climbing Yesler from 14th to 18th seems to just crawl along, but before I know it I'm at the top, and not all that tired.  And I'm out of shape, 5'9"/230lbs.  

Check out alternative routes. You don't have to love climbing hills. Figure out which kind of hill you like more, short and steep, long and fast. Then, try to find different routes that suit you better. And remember, when we were kids just riding around was nice too. I find my camera helps me take a break to continue enjoying a commute I do often.

If you are on a multi-speed bike you should with a hard gear to build you speed and momentum. As you hit the bottom of the hill shift down to an easier gear to keep the same cadence, RPM, and stay seated. It takes less energy during extended hills if you are seated.

If you are on a single speed bike you need to start by standing and giving it everything you've got as you come to the hill, and then sit back down and take advantage of your momentum as long as you can to rest for you next burst of weight assisted pedaling.

The best way to improve hill-climbing is to climb some big hills! I'd recommend Cougar Mountain on the Eastside. You can search for a route or join one of the Cascade rides.

Depending on the route, it's about 35 miles round trip. Ride that once every couple of weeks and you'll be climbing hills with the best of 'em.

I too hate hills, love my bike, and live in Seattle. Here's how I get around it:
1) Find something so cool at the top of the hill that I must get there. If what's at the top of the hill is just the top of the hill, that isn't much incentive.
2) Go up in stages. I may not make it all the way up before I have to walk, but maybe I can make it 1/4 of the way. With more practice, I do 1/2. Eventually I will conquer the steep hill at the end of one destination because I can see myself progressing.
3) Find a parallel, but less steep, hill. Seattle slants in all kinds of places; the W>E hills north of downtown (for example) are shallower than those south of downtown.
4) As Fnarf says, shift. Lower gears really do help.

Car hook.

in short, replace repetition with variation, and compete with yourself.

Lots of folks have talked about attitude. Mine is simply: good/asskicking hills are good challenges that might very well be opportunities to show yourself the kind of character you have.

As far as your specific Q of improving hill-climbing abilities...
ingredients:
A good/well-maintained multispeed bike - lightweight, well-lubed, with easy gear changing. No fixie schwinn from the 70's if you're serious about improving your hillclimb ability.
A good fit: a bike that doesn't fit or isn't sized to you correctly will make your hill climbs exponentially harder.
Know your best stance: some people climb hills better seated, a few climb better while standing on the pedals. Figure out which works for you, and more importantly figure out -by trial and error- which certain stances work better for certain types of hills.

but above all: Train & condition your body.

I used to hate hills but it took all of 1 week of riding around the county to realize the hills are everywhere and aren't going to change anytime soon.

I always feel that the downhill aspect of a hilly roundtrip is the reward, so I would choose routes with multiple ups and downs (lots of little downhill 'rewards') rather than one long slog. As it happens, this is an excellent way to exercise your leg muscles too, and builds up your body's ability to climb the bigger monsters.
An excellent less-than 50 miles training route is Coal Creek parkway - a sort of roller coaster for bicyclists. For a longer training route: I-90 (there's nothing quite as satisfying as cresting the pass summit of the Cascade Mountains on your bike). Once your strength/hill climbing muscles are growing, you might start doing things like going out of your way to hit the bigger/nastier hills in town (Marion/Madison from waterfront to First Hill; N. 65th St crosstown; Denny Way, or *choke* the QA Counterbalance).

Use foot straps!

And a swig of beer before and after won't hurt either. Also, always take advantage of the view once you get to the top. There are some beautiful vistas in this city no doubt.

Clipless pedals. (paradoxically, this means the ones that have the shoes that snap in, if you didn't know)
They make hills happier. If no clipless pedals? Tighten your cages so that you are getting some power from pulling your legs up. If no cages? Buy cages and straps. They will make every bike ride better than with flat pedals alone.

Standing up to climb helps use different parts of your legs which many find either more comfortable, or able to provide variation in movement that keeps a longer hill interesting and wards off fatigue. It also allows you to pull the bars toward you differently, which gives you leverage pushing the pedals down. Also, lean your body forward from the seat a bit to move your center of gravity closer to the stem. Racer types are out of the saddle quite a bit on climbs for these reasons and probably more science-y ones, too.

Also: a weird thing that helps when I ride with clipless pedals or flat ones standing up while riding up hills is focusing my mind on the ease of pulling my legs up and letting my weight push the pedals down. It is "bad form" because it kind of makes you "chop" at the pedals with the down stroke, but if you're tired as hell, I say do whatever is easiest and comfiest.

Do leg workouts with weights if you can use a gym.

Conquer riding up Denny. Then everything else will seem pretty easy.

If that is too much to take on then learn to zig zag. IE: Go up a block then cut over, then go up again.

Learning the alternatives routes is important. Sometimes by just going down a street or two the grade of the hill changes allot.

Learning proper shifting technique is essential. Make sure to shift down before the hill becomes to tough.

But when all is said in done, I believe one needs to just toughen up a bit and embrace it. Remember what a bad ass you are becoming and strong your lungs are getting by doing this.

One thing that has always worked for me is a tip from my mountain biking friends regarding seat position. Basically, make sure that your seat is at the height where, when your ass is seated, a full extension of your leg results in the back of your foot sitting flush against the flat portion of your pedal. Saved my ass on 2K logging road climbs, and I see alot of folks whose seats look to low around here.