Can I receive child support from my ex's weekly unemployment check?

Yes.  Unemployment benefits are still income, for child support purposes.  It's possible that the court might reduce his weekly amount, based on what he's getting, since it's relatively easy in this economy to persuade a court that he's looking for work but can't find it.  I don't know whether or not you can get it taken directly out of his money before he gets it, like a wage garnishment, but if you're in court about this, ask for that, the worst that happens is they say no.