I've had mixed results with RIT dye on polyester or poly blends, but particularly with dark blue, you have to keep the water hot or it'll start going purple rather than blue. Generally with any dyeing you want to stir the fabric every so often, say every 10-20 minutes, to make sure it's getting evenly covered, no bits folded in on itself, etc. After an hour or two the color will pretty much be what it'll be (except that it'll keep going more purple as the water continues to cool), so no point in leaving it overnight. If it's only light purple after about 30 minutes, the polyester might not be absorbing the dye at all. Careful washing it out, as it may just be sitting on the top of the fabric, so anything you wash with it will also turn blue/purple.
If you can find it, there's another dye brand called iDye that I've had a lot of luck with. It comes in regular iDye for natural fibers, and iDyePoly for polyester, and is the only thing I've found that will dye 100% polyester. You can use both together for blends, too. The dye is designed to go in your washing machine (and I haven't had much staining, even after repeated rounds of dyeing), but for blue if there's any way you can use the stove-top method instead, you'll get a much better blue, less purpley. With a trenchcoat you'd need a pretty big stock pot to dye in (and you can't use it again for food, so a cheap or second hand one is best), but you'll get the most consistent and true blue that way.
