Had the same issue myself. My Wii stopped playin' dual layered discs a few years back; considerin' Brawl is one, I wasn't able to play it 'til I got my Wii U recently. Since the Wii U ain't got 'cube controller ports, I'm stuck with the Classic Controller. =/ Felt awkward as hell for the first few months, let me tell ya.
Thanks Mr. MR and ill add you splash in a minute, the few experiences i've had online is that it lags a lot when it comes to random 4 player online, when its only 2 its a bit better, my connection is at 3 Mbit/s sometimes and sometimes it lags or i get disconnected I just hope it goes quite well