The most successful invasion was probably that of the Ulrican tribes into the Reik basin in -1000 or thereabouts. Or possibly the orc invasion from space.
Leaving those aside on the grounds of facetiousness, I'd make a case for the Skaven in the Empire in 1111-1124(ish). They came very close to conquering the whole thing, and it was only really bad luck that stopped them (who'd have thought the Undead would interfere?)
Ushoran's attack on the pre-unification human tribes under Neferata seems to have been an unqualified success. Unfortunately the orcs stopped it from having any lasting significance. Nagash's attacks on Nehekhara are debatably successful. He did succeed in wiping it out altogether, but that wasn't really by invasion (the actual invasion failed) and he cocked up the aftermath so didn't get to hang onto it.
If you believe that Grom really sacked Middenheim (which I don't; it's goblin propaganda) then he's probably the most successful greenskin warlord. Other than that, Gorbad. According to Legend of Sigmar (again, eurgh) Cormac Bloodaxe made a reasonable fist of things - killing two Counts and destroying most of the north a la Archaon, and Nagash did even better.
It depends really on what's meant by success, too - most ground covered or the totality of victory achieved, or some combination of the two, or some other factor. Gorfang Rotgut's raid on Karak Azul was an unqualified success, and Skarsnik has been almost entirely untroubled since taking Karak Eight Peaks, but both of those were fairly modest achievements in themselves.