Is the cat on anti-biotics? I've had a cat vomit from anti-biotics before. From what I've heard, it's common but you should either try a stomach protecter medication along with the anti-biotics, or stop giving the anti-biotics.
It's possible the vomit is just from the cat eating plant matter, but I think you'd be able to tell if that was the case. You'd see bits of plant or grass in the vomit.
I think the bile sounds pretty likely. I hope it's not too serious, as the cat that I mentioned above was really quite ill at the time of giving the anti-biotics, and vomited large amounts of green liquid again before she died.
Best of luck, and I hope your kitty is ok!