While you can paint warps while on the loom, that sounds too messy for me.
This is a method that I learned while taking a workshop with Eileen Hett and Mary Berg. The way I painted my tencel warp using Procion MX dyes was as follows:
Wind warp. Tie cross firmly, but try and use a different type of fibre from your warp yarn (eg use protein fibre if warp is cellulose fibre). My warp was short so I didn’t use any choke ties. For a longer warp I would have made figure 8 ties about every 2 yards or so to prevent major tangling.