How do you improve my climbing endurance?

I want to improve my climbing endurance whether road or mountain biking. I'm referring to multi-mile double-digit grade climbs. Since I don't have easy access to these trails, how can I train in the city for these weekend rides?

This is a really tough question. Without a doubt the best way to train for climbs, is to ride a lot of climbs. The next best thing is simply to work on good general conditioning. It's hard to hide poor conditioning on a climb. As for specific training for climbs, which doesn't entail going up climbs, I can give you a couple of suggestions, but I really don't know how effective they will be. First, find the closest thing to a long climb that you can find in town, put it in your biggest gear, sit down, and grind your way up the hill. You want it to be as difficult as possible to keep the pedals turning over. This will help your body learn to pedal a complete 360 degree circle. Your body doesn't naturally want to pedal a 360 degree circle. It has to be trained to do this. If your RPM's drop below 20, that's great. You want to be able to focus on pedaling a complete circle, actually you want to be forced to pedal a complete circle. Hopefully, after a while your body will become accustomed to pedaling a complete circle, and will just naturally shift to full circle mode whenever you hit a tough stretch of climbing on your weekend rides. This should help immensely.

The second thing that you need to work on is the exact opposite of the first, you need to do some high RPM work. The first exercise is simply to teach you to pedal a complete circle, but you do not want to grind up hills in a big gear during an actual ride. You want to hold fairly good RPM's. Generally this exercise is best done on a nice long climb, where you can put it in as small of a gear as possible and keep your RPM's as high as possible for the entire climb. If you don't have a long climb in town, then the next best thing is probably to try to do a one hour ride, once a week, where you try to keep your RPM's above 120 for one hour. It's always a good thing to work on keeping good RPM's, but once a week set aside one hour to work specifically on that.

These two things, pedaling a complete circle, and holding good RPM's, along with better conditioning, should make you a better climber.

Timon is right about the best way being to climb.

Some of the cycling trainers have climbing blocks where you can your front wheel up in the climbing attitude, then add resistance until it builds.

Some studies have shown Intervals (sprints) to benefit overall endurance, and it certainly will help build the muscle fibers.

Or try a stairmaster or stepper at the gym--those things kick my ****.

Try riding a crappy old steel mtb with nobby tires on it uphill on the pavement.Get yourself to where you do not use the granny gear[smallest front ring].
You are now a climbing powerhouse on your trail bike.