Don't put them in the oven. There's no baking supposed to be going on. You're pan-frying the meat.
Heat the grill pan to moderately hot. You'll want to see grill marks in the finished patties.
You're not making meatballs here. Do not put a ball in the grill pan and smash it down into a patty.
You'll want to avoid handling the patties as much as possible to avoid making them tough. Grab a handful of ground meat and gently flatten it out in your hands.
Making the center slightly (.25 in./a few mm) thinner than the periphery will keep the patties from swelling up in the middle.
Don't add any oil. If you use an 80/20% or 85/15% grind, that's a lot of fat to begin with. A higher fat content is not recommended. You could add a pat of butter--mostly for flavor.
You can season the patties before you cook them, but the "usual" way is to add salt/pepper in the pan while the opposite side cooks.
Gently place the patties in the pan and cook over medium-high to high heat for about three or four minutes. Flip the patties over and cook for about 3 more minutes.
That's all. No baking, no oven, no broiling. Just pan-fry on each side for around 3 minutes. The grill ridges on the pan should elevate the meat enough while it's cooking to drain off most of the fat, but you want a little bit to remain for flavor/moisture, etc.
For absolute optimal results use an outdoor grill (barbeque) with good charcoal. Pan-frying is only a "second-best" option.
Here is a very comprehensive tutorial: How To Make Burgers on the Stovetop Check out the pictures for details on making the patty, the color (fat content) of the ground meat, etc.