You can play with a number of different controllers in the Wii U version.

Wii U GamePad

If you are playing alone, you can play using only the Wii U GamePad screen.

Wii Remote Plus

You can play by holding it sideways. You can use a normal Wii Remote as well.

Wii Remote Plus + Nunchuk

Wii Remote Plus + Classic Controller Pro

The game supports the Classic Controller as well.

Wii U Pro Controller

Nintendo GameCube Controller

You can connect up to four GameCube Controllers using one GameCube Controller Adapter. Connect the adapter to the Wii U console using two USB cables. (You can use the USB ports on the front or the back of the Wii U.)

Nintendo 3DS series systems

You can use the Nintendo 3DS family of systems as controllers as long as you have a Super Smash Bros. for Nintendo 3DS game for each system.
If you don't have the Super Smash Bros. for Nintendo 3DS software, you can purchase and download the Smash Controller software for Nintendo 3DS beginning 6/14/15 (eShop price: $0.99).
(Supported only in Smash mode.)