Ordinarily, I don't recommend painting vinyl and linoleum flooring. The surfaces don't accept paint well and are likely to be dirty, stained, or waxed, which further inhibits paint bonding. And painting won't hide holes or dents.
If at all possible, tear up old flooring and replace it. But if you absolutely have to have a quick fix for a single event, for instance, and aren't worried about long-term durability, you can dramatically change a room's look.
Before painting vinyl that's in sound condition, test the paint on an inconspicuous spot to make sure it will look okay when complete.
Sand the vinyl with 220-grit sandpaper to dull the shine.
Wipe clean and then apply a liquid deglosser to improve bonding. Let dry.
Prime the floor, then roll or brush on one or two finish coats.