Go through the level Oil Drum Alley. Get Rambi who's in a hole by the middle gate... you can spot the hole by a single banana that you jump to. Then, go backwards a bit before the middle gate and you'll find a blast barrel below you resting on some ground. Leave Rambi behind and as Donkey Kong, try jumping down next to the barrel. It may take a couple of tries for you to get next to the barrel without going inside of it. If Diddy is still on the ledge above, just jump. Now, this is difficult. it takes many tries. Hit select twice in a row, but don't keep on jamming on the button. If done correctly, you will switch twice and there will be two little bursts in the barrel. Land on the area to the right. If done right, you'll be able to control both Diddy and Donkey Kong. If you do this with Diddy in the beginning, the game will freeze. When you are controlling them both, each can get injured. If one falls into a hole, then you will lose a life. Often Diddy will go farther than Donkey, so to even it out, you can push down and then roll diddy in different directions by pushing Y and left or right (while pushing down).
Now, make Diddy jump onto Rambi. Then, make Donkey jump onto Rambi. The colors will glitch up - Rambi will double, or there may be two Donkeys or Diddys, or one of them may be floating in thin air. Try pressing A or injuring one of the characters and you'll get all sorts of weird outcomes. Sometimes you will even have a badguy attached to you. Diddy once made a badguy appear out of nowhere and hit Diddy. Try experimenting!
A Youtube video of this glitch can be seen here.