1) fun as a barrel of monkeys = really fun, but usually use it sarcastically which means that explaining something that is not at all fun.
example) I'm going to the dentist to have a cavity filled. Fun as a barrel of monkeys.
2) Monkey business = silliness, dishonest behavior
example) You might say to a room full of rowdy kids, 'Enough with the monkey business'
3) Monkey suit = tuxedo or other formal evening wear for man
4) Monkey on my back = This is a problem or something that's really stressful that's taking a long time to resolve or won't go away.
example) My friend is going through a nasty divorce. It's a monkey on his back.
5) Go ape = get really angry.
example) If she screws up my car, I'm going to go ape.