If you want to paint, you'll need to key the surface so scouring pads or similar, etching primer and top coat. Still, you're painting on top of chrome so if that starts to flake off your paint is coming with it.
Plasti-dip, as Shite Knight has found with his Fresian Sierra, only really sticks to itself - but on a luggage rack that's ideal. Give the whole thing one good coat and it doesn't matter if it sticks to the chrome or not, it'll form a luggage-rack shaped outer shell.
You can also look at "bed liner" or whatever which is a hardcore version of plasti-dip intended for pickup beds.