Colleague stole my work and is publishing it as his own; my PhD supervisor supports him; I got the paper to review. How to proceed?

You shouldn't even think about it, do not review that paper, it's a clear conflict of interest, even more because the author is your direct colleague. And the only solution is to sit all together and explain to them that your contribution to said article is significant and you would like to be a co-author. If they have problems with you being a co-author and you have done substantial work then I would seriously consider if you want to be in this research group.

There are two different questions here. One is easy, one is hard.

The easy question is: "Should I review the paper?". No. Definitely not. It goes against the ethical standards of every respectable journal I can think of, and you should contact the editor immediately.

The hard question is: "What should I do about the authors not recognizing my contribution?". It sounds like you have already approached them regarding this. You should try this again, bring a copy of your old poster, and clearly state that your goal is to be a co-author of the paper. If you are unsuccessful, you can try to approach a department head or similar, but that will most likely not give you anything but grief. If you think it would be possible to actually publish a similar paper yourself, you can of course do that. In my field it would not be possible to do that, but I guess it varies. At least you will have learned the valuable lesson to write up and publish your stuff if you want to be recognized for it.

I see two possibilities here.

1. The poster doesn't prove decisively you must be credited for the developing the method

In this case, you are in quite a bind. You see, nothing you've told us can be established as fact, as it lacks corroborating witnesses; it will be your word against your colleague (and perhaps even the word of your advisor), since nobody is ever going to examine the details (like source commit logs) to determine who's right. The situation is indistinguishable from the outside, without much effort, from you just trying to claim credit that you are not due.

I would suggest, in this case, to write the PC chair, inform him/her that you are certain this is a paper by your colleague, and that you can't review the paper since you're biased - without saying a single word about the ethical issue, or why and how you're biased. Then go tell your colleague that you've done so - but show him a printed copy of the email, don't send it beforehand, so that s/he doesn't misconstrue anything and try to use it against you somehow. When you talk to him/her, try to make use of the fact that you did not take the opportunity to claim right-of-authorship vis-a-vis the PC, as psychological leverage against him: "I didn't tell them that XYZ because I think the fair thing is that we resolve this issue between us, but how could you submit a paper behind my back like this? etc. etc." You might even go as far as threatening to claim his paper is taking credit of your own work, as a way of pressure him/her: Even though you won't do it, it's still the danger for him/her of a lot of hassle and a question mark on his/her reputation. You could also threaten going to the faculty management, or whoever is in charge of graduate studies (i.e. not through your advisor), or taking it up with the graduate researchers' union for an internal resolution dispute (I hope you have such a union) etc.

Note: I'm suggesting this somewhat aggressive behavior since your advisor won't back you. If he had been at least neutral I might have suggested something different involving him/her.

2. The poster proves you must be credited for developing the method

This next piece of advice is, again, due to the fact that your advisor has threatened you and has acted in a generally underhanded way; had that not been the case I'd make another suggestion that would be more forthcoming and open.

Anyway, in this case, write the PC chair. Inform him/her that you are certain this is a paper by your colleague; and explain how you can't review the paper since you're biased. But write the explanation so that it is clear that you did a lot/most of the work - without suggesting you should be listed as an author, but in a way in which any reasonable academic would think that you should. Most importantly - refer the PC quite early on to the poster and relate your explanation to what appears on it, to lend your description more credibility. If there's some publicly-available source code which has not been covered by the poster, of which you are the author - refer to that as well.