When does one "become" a philosopher?
Right after the secret handshake...
More seriously, there's no single answer, and no clear one in any case. Does someone who has a BA in philosophy count as a philosopher? How about someone who has no formal education in philosophy, but through lots of reading and informal conversation has gotten good at the sorts of things philosophers do?
Sufficient conditions are not so hard to come by. Someone who regularly publishes in recongnized philosophy journals would count. So would anyone who regularly teaches bread-and-butter philosophy courses in a university philosophy department. But not all philosophers publish, and not all teach.
Necessary conditions aren't hard to come by either. Someone who had poor verbal skills and neither has nor ever had any talent for thinking analytically wouldn't count as a philosopher. But there are many linguistically gifted analytically-skilled thinkers who aren't philosophers. (An excellent lawyer, for example.)
There are some useful generalizations, of course. Philosophers typically have studied philosophy in university. In any case, they generally have a wide acquaintance with and a good understanding of philosophical texts and ideas. They are interested in certain sorts of questions, though there's a fair range of possible interests and aversions here. (One philosopher might be fascinated by questions about the foundations of morality but have no taste for theory of knowledge; another might have more or less the opposite preferences.)
The most promising answer may seem like a cheat at first: you become a philosopher when you've reached the point where philosophers would agree that you are a philosopher. In one way, this begs the question: it assumes without giving a deeper story that there are already some people who count as philosophers, and it bootstraps from there. But there's no cure for that; "philosopher" isn't a natural kind, marked out by some reasonably tight set of defining characteristics. There are, however, many paradigm examples. (All members of this panel, for instance, are philosophers by any reasonable accounting.) And there are reasonably well-defined formal and informal practices, networks, associations and the like that philosophers recognize and that recognize philosophers, as it were. (Publishing in certain journals, belonging to certain associations, having certain educational backgrounds, working on certain problems...) On the one hand, all this is still fuzzy around the edges. (For example: there are academics who have all their training outside of philosophy and aren't employed by philosophy departments, but who have made valuable contributions to the field. Are they philosophers or not? There's no sharp answer.) But on the other hand, there are people who have an amateur and sometimes crankish interest in philosophy and who think of themselves as philosophers. Most of the people who count as philosophers by all reasonable measures would disagree. Surely that should carry some weight!
The suggestion here is hardly original, and would work in a good many other cases: mathematician, historian, chocolatier, chef, musician... I might be able to knock together a few pieces of wood to make a bookcase. The guy who's rebuilding my porch, however, wouldn't count me as a carpenter, and since he clearly is a carpenter, his opinion is worth a lot more than mine.
So when do you become a philosopher? Not at any one moment, but gradually, as you reach the point where philosophers count you as one of their own.