Just thinking aloud-soaking it would seem a more practical way but would whatever you were soaking it in seep through to the bottom of a tightly wound spool of new line? I have noticed when I run my line through a wad of mud for sinking purposes how it dirties and takes the shine of line. Could it be as simple as that, soaking it in a dirty solution of mud and water for 24 hours bearing in mind the first point.
My second thought was one of those cheap dyes you can buy, I think some of them are just water based so maybe they could be diluted to the required level for this purpose. As I say, never tried it before, just thinking aloud. Maybe a bit of experimentation in jars with some small pieces first might help.