You need to use acid dyes on nylon. My preferred acid dye is Jacquard Acid Dyes. The "acid" involved is not caustic; it's just vinegar. Vinegar can leave the final outcome smelling kind of funky though, so if you don't want to leave your tights in a room with an open window for 3 days to air out, you can substitute the vinegar with citric acid crystals.
Food dye is pretty likely to wash out and fade...and it'll bleed when you sweat. It will come off on you and your costume, and will probably leave sweat stains behind. You might want to wash the hell out of your tights to try and get as much of that stuff OUT as you can.
Low-end off-the-shelf dyes like Rit and Dylon are catch-all dyes that are mostly formulated toward cotton. (And they kind of suck.) Dyes for cellulose fabrics don't take to nylon.
I get my dyestuffs from this site, and they have great instructions: