We inherited vinyl flooring in our kitchen and I mop it, or if its really mucky, get down on hands and knees and do it with a cloth.
One thing I don't advise doing is scrubbing it with wet sandpaper to get paint off it. But then I think only a stupid, stupid man would do that and not realise it would wreck the surface. Still, if you are decorating your kitchen onabudget why not throw in a new floor as well? Men
LOL...er.....not so silly woman who did put the nice new vinyl down 2 hours after glossing the skirting boards used silver scrubby thing to get wee patches off fresh white gloss off of it....gently though...no harm done!!!!! women!!!!!!!!!!