Just go buy a black sharpie (the one that is metal, and looks like a paint marker - and some clear coat.
Remove the back cover, and sharpie over the logo, and the rest (want the color to match don't you.)
After that let it dry for an hour, then coat it with three coats of clear with a 30 minute gap between the coats to allow for drying.
BEfore proceeding, let it sit for anywhere between 6-24 hours just to make sure that everything's "Dry."
When done doing that, set the oven to 150 (bake). Let it heat up, then when it's ready turn it off, and let it sit for a minute
Put your piece in DIRECTLY on the rack, OR on a backing sheet (tin bakery pieces are useful for this.
Leave the piece in the oven for anywhere between ten minutes to five hours depending on how you want the paint to "cure." The longer it's left in the longer it'll last.
It's a smart idea to test a piece of plastic that's about the same consistency as the one you're going to do this with, so that you have a good idea of when it may start to get just a little too close to melting. Some plastics can take a constant 150, some can't. it all depends on how it's made.
It's worthwhile to mention that you want to open your windows while doing this, as first the paint will "de-gas" and it WILL stink. Also, the oven won't be "food safe" for a good two days after this, so when you're done, you'll have to leave it open (off - but open) and if it's a "self cleaning" do NOT DO NOT DO NOT run this cycle. Self cleaning ovens actually have a chance of exploding if you have put too much paint gas in them, and run the cleaning cycle.
You've been warned.
Enjoy.
You may have to do this like once every 6-12 months depending on how much you actually USE the phone, as the oil will dry out the clear coat, but all you'll have to do is recoat the clear.
I've used that exact methods to color the plastic parts on computer cases, and miscellaneous other electronics for the last 14 years and I've never had anything melt, so don't worry.