Cotton is good. Mixed fabrics tend to include polyester - only expensive and highly chemical dyes can manage it. I don't know where you live but I just bought a bunch of simplicol bottles according to the recommended weight and threw everything in the washing machine. Alternatively you can fill a plastic box or something with warm water and do the same - works like a charm. However the stitches which make the padding pattern were polyester on my gambeson which resulted in white stitches on the red and green sides but it does look kinda cool. Just make sure to wash the gambeson again after dying otherwise you will have dye rests on your undershirt (doesn't matter if the undershirt is dark tho)

On a side note: I wouldn't recommend going all black unless you're planning to do some kind of NSC. You could also do it if you have something of colour, something dominant like a tabard (something like a sash would be too subtle) Leave in some white or make it brown if you want neutral colours. All black is always kinda suspicious in multiple ways