What the other answers have said about "which" having to be used with commas (or in non-restrictive clauses) is wrong. "Which" has long been used by respectable writers in restrictive clauses as well. And when it's used in a restrictive clause, it's wrong to use a comma before it. There's a separate question about this: When to use “that” and when to use “which”?
Consider the bear which scratches his head.
This sentence does sound wrong, but for another reason. "Which," whether used restrictively or nonrestrictively, has a strong tendency to be used with inanimates. For this reason, "which... his" sounds bad, because "which" implies you're thinking of the bear as an object, or at least as a not-very-animate thing, while "his" implies you're thinking of the bear as a person, or at least as a somewhat animate being. "Consider the bear who scratches his head" would be better, or, as the other answers mentioned, you could use "that" and say "Consider the bear that scratches his head." (In general, "that" is not as common as "who" when referring to people, but either is grammatical.)
ShreevatsaR in a comment mentioned a third option, "the bear which scratches its head." This doesn't sound as clashingly bad to me as "which scratches his head," but it also doesn't sound as good to me as any of the alternatives I listed in the previous paragraph.