Probably none of them without a mouse with a bunch of extra buttons. This game wasn't intended to be played with just a mouse. All of the champions are designed around having all of their abilities bound to a key. It would be pretty awful to try and click the spells as you use them, you'd get outplayed every time. Even being really slow on a keyboard would be better than trying to individually click the spells.
Indeed no champion is ment to be played with a mouse only, so you wouldn't stand a change against other players..
However a good champion I can think of for this would be Sion. (Especially when going Attack damage on him).
You can activate the ult or shield etc without having to interrupt the autoattacks so I think he fits that playstyle best.