Five Monkey Trial is like a series of five sequential bonus stages. You must collect all of the banana coins in each room to proceed to the next. There are no DK Barrels in this level, so get Diddy Kong beforehand if you want to make the stage, especially the final challenge, a bit easier.
First, proceed to the right and pound on the DK platform to make a barrel appear. Jump in and you'll be at the first challenge. Collect all five banana coins, using a combination of low jumps and high jumps to avoid the enemies. After collecting all five coins, grab the puzzle piece that appears in the center, and jump in the barrel when it appears.
In the second room, you'll have to dodge falling pillars. Watch the background and keep jumping from space to space, collecting the banana coins that appear after the pillars fall. Once you collect all five, you can grab the next puzzle piece and move onward.
The third challenge has you jumping on a series of squids to reach the top. Wait until the first squid is safe to jump on, then jump off the spring and high-jump off four squids in a row. After reaching the top, collect all five banana coins and the puzzle piece when it appears.
In the fourth room, you'll need to jump over enemies and collect banana coins on a rotating platform. There will be blocks moving above you, so be sure not to trap yourself below one of them when an enemy comes by. Move to the right if you need to, but be careful not to fall off the platform. At the end, a barrel and a puzzle piece will appear.
Finally, you must dodge falling explosives while jumping between two platforms. Watch the background and keep moving to safe areas. After each set of explosives, jump between the platforms to grab a banana coin. The fifth set is the hardest; you must jump between two rows of explosives right before they hit the ground, and there is very little margin for error. Diddy Kong's hover ability makes this easier, but it is possible to do with Donkey Kong only. Collect the last banana coin and puzzle piece, and jump in the barrel. Jump up to the orb to end the level.