Diagnosis
You didn't do anything wrong; this is the infamous 'grease waterfall' issue, especially on early cooks or with fatty meats. The grease is missing the collection chute and running down the inside of the back panel.
How to Fix It
The primary fix is to ensure the grease management ramp is installed correctly and is *level*. Turn the grill off and cool. Remove the flavorizer bars and heat diffusers. The metal ramp that should direct grease to the chute can get misaligned. Bend it gently so it sits perfectly flat and leads directly into the hole for the grease chute. Also, make sure the rubber gasket on the grease chute itself is present and seated.
If That Didn't Work
If the ramp is aligned but grease is still leaking, the grill might not be level front-to-back. Use a spirit level on the cooking grates. The grill needs a very slight tilt backwards so gravity pulls grease toward the chute. Adjust the wheels or legs until you get that slight rearward tilt.
Heads Up
Never let the disposable grease cup overflow. It will wick grease back up the chute and make the internal leakage worse. Check it every cook, and replace it before it's more than half full. A full cup is a major fire hazard.