Shouldn't take too long, just expect to lose a fair few matches. Here's how I did it:
Pick Bison to begin with, and spam psycho crushers for the first few fights. Your opponents won't be too aggressive in the early stages so you can easily get away with this. After your first 2-3 fights, things will get a little tricky depending what order you have to fight the remaining characters in. I've given strategies below on how to counter each character.
Ken - Can be a pain if you happen to get him right before the final four. If you don't fancy fighting him legit with someone like Ryu, pick Bison again and spam the p. crushers. It's a little harder to do with Ken than other characters, mainly because he'll dragon punch with perfect timing after a few crushers. If he does this, which he most likely will, back off and block his fireballs if he decides to throw some, then use p. crusher the first chance you get. Repeat.
Ryu - Same as above, bit easier considering he doesn't dragon punch out of the crushers like Ken does. Be patient and block his fireballs, then wait for him to either jump or move towards you before launching a p. crusher.
Guile - A lot easier to fight if you get him in the early stages with the Bison method. Still works later on but will take a few attempts. He won't sonic boom as much as you'd expect.
Chun-Li - May take a few attempts but p. crushers work perfectly. If that won't work, try beating her legit with Ryu as it isn't too difficult, depending how late on you have to fight her.
E.Honda - Either fight legit with Ryu or spam the crushers with Bison, both work fine. Easier with Bison, just don't get cornered or you're most likely getting hundred-hand slapped.
Blanka - Not really a problem, but if for whatever reason the p. crusher tactic won't work then pick Blanka and spam jumping h. kick. With good timing, he won't be able to land a hit.
Zangief - Actually took me several attempts, he was my last fight before the final four. The p. crusher tactic works fine but he's also easily beatable with Ryu. Try not to get trapped in a corner, easier said than done.
Dhalsim - Possibly the easiest character to beat, p. crusher all the way. If he teleports, it'll most likely be towards you so slide kick as soon as he reappears. If you choose Ryu like I often do, keep your distance and fireball. If he teleports, low h. kick.
Balrog - Pick Blanka and spam electricity. He won't land a single hit on you. Bison struggles here.
Vega - Again, use Blanka and spam jumping h. kick. This should land you 3-4 hits before he gets wise to it and starts using his crawling-up-the-cage tactic. At this point, avoid him like hell by jumping to the opposite side of the screen, and try to land a jumping h. kick on him when he leaps towards you.
Sagat - Takes a few tries but again, p. crusher will beat him. What usually happens is, you launch a p.crusher which he'll most likely block, soon as you land he'll launch a high fireball which gives enough time to hold down-back on your analog, duck the shot and launch another p. crusher. Repeat, even if he blocks he'll take fairly hefty damage.
Bison - By far the most difficult character. One of the solutions on here suggests the Ryu method of spamming l. punch, but this is practically impossible if you don't have a turbo controller like me. (I've tried doing this by hand, it hurts and as soon as you stop spamming he'll land a heavy kick on you). If you don't have a turbo, pick Bison once again and use the p. crushers but expect to lose several times. He'll throw you from seemingly absurd distances at times, and if you're backed into a corner he'll be all over you. Just keep up the p. crushers until you beat him, and try not to launch them too close to him or he'll block and immediately m/h kick you before you even land and turn around to him. Another option here, suggested by se7en sin, is to pick E.Honda and simply spam the X button to keep him at bay. Whichever method you choose, stick to it and you will eventually prevail.
TIP: To conclude, every character is beatable with either Bison or Blanka, but something to bear in mind: as mentioned in my strategy for Bison, be wary of when you launch your p. crushers. The best place to do them is when you're closer to your side of the screen but not trapped, and your opponent is approaching. This way, you'll land clear of him/her regardless, unless you're punched out of it which is unlikely. If they block, you'll deal damage and land completely clear of them on the other side. I mention this because, if you launch one closer to their side of the screen and it gets blocked, you're in danger of being hit, trapped or thrown once you get back up. Good luck, and don't get too frustrated... Street Fighter games are notorious for this.