ALKALINE based cleaning agents will clean up grease nicely but some Tri- Sodium Phosphate mixed in hot water will remove the grease, as well as the paint if left long enough.. LOL.. Sorry couldn't help myself.. – NOTE – DO NOT DO THIS, I’m being silly, this stuff can remove the skin from your hands, as it burns..!!
Mr Muscle oven cleaner can remove "some" dirt and is a whole lot safer, but if it's really bad you might be better off getting a company in that has chemicals that'll clean it up for you. Worth doing IMO, as after all, this is where you are cooking food. Even though the bacteria is subjected to high heat and mostly killed off, it doesn't look too appetising when you open the oven door and take out your freshly cooked frozen pizza.. Yuck..!!
Oh - Easy off and Mr Muscle can also strip off water based acrylic paints, and damages enamels as well, pretty powerful and nasty stuff actually... Wear gloves..