For the most part, you can have confidence that homemade hamburgers you prepare from scratch with ground beef are gluten-free.
Pure beef is safe, and any hamburgers you make out of 100% ground beef also will be gluten-free, assuming you don't subject them to gluten cross-contamination when you're cooking them.
You should be able to find a gluten-free bun to put that homemade burger in, too plenty of gluten-free manufacturers now make them. (I provide the list in Gluten-Free Hamburger and Hot Dog Buns.)
Some pre-formed hamburger patties you buy at the store do contain grains (almost always wheat) as fillers. If this is the case, though, the manufacturer wouldn't be able to claim the burgers were "100% beef," and also would be required to disclose the filler in the ingredients list. If in doubt, you should be able to call the manufacturer's toll-free number and ask.
Also, you should never eat a burger (or anything else) prepared by a friend or relative this article: Gluten-Free Food Prepared by Friends or Relatives explains why.
Still, it's in restaurants which aren't required to disclose ingredients where you can run into the most trouble with gluten in hamburger meat.
Many restaurants serve pre-formed burger patties that include additional ingredients, including wheat (again, as a filler). Others make their own burgers in-house, but use bread crumbs to stretch the meat and improve the texture of the burger.
You won't know about this practice unless you ask ... and when you ask, you need to question someone who actually knows how this works at that particular restaurant. Generally, that's either the chef or the manager, not your server. (That's just one of my tips for dining out: Five Rules to Stay Gluten-Free at Restaurants.)
Most fast food restaurants seem to serve hamburgers that don't include gluten grains as an ingredient. In fact, some people with celiac disease and gluten sensitivity say they have good luck ordering bun-less burgers at chains like Wendy's, In-N-Out Burger and Sonic Drive-In. (Check out my overall article on gluten-free fast food for the menu details.)
Of course, when ordering fast food, you need to beware of the potential for cross-contamination make sure the workers change gloves before serving you, and that your hamburger never goes near the gluten-y buns.
So the bottom line is, you shouldn't have too much trouble finding a gluten-free hamburger, especially if you're careful to read labels on pre-made beef patties and to talk to the chef at any restaurants. Enjoy!