Diagnosis
The leak is almost always at the connection between the chute from the barrel and the cup itself. The cup isn't sealing, either because the gasket is missing/gunked up or the cup isn't seated right.
How to Fix It
Take the cup off. There's a small silicone or rubber gasket that should be inside the groove of the cup's opening. Clean it or replace it if it's cracked. When you slide the cup back on, give it a firm upward *push* at the end to click it into the locked position. You should feel it engage.
If That Didn't Work
The chute itself might be bent or warped from heat. Inspect the metal lip. If it's deformed, you can sometimes gently bend it back to shape with pliers to create a better seal.
Heads Up
Never let that cup fill more than halfway, especially with hot grease. The weight and heat can warp the plastic and ruin the seal permanently.