Try Joe's approach first as far as cleaning. He is also right. You want to use a high heat enamel if you try to repaint. You can get it in rattle cans. For example but not to hype Rustoleum although it is a good brand.
Your auto parts store will have high heat engine paints and perhaps a better selection of sheens.
It will never be the same as what the factory put on though so also as Joe points out, do not count on being able to use things like oven cleaners again.
You will want to prep as you would any job.
I would be a bit concerned that you are experiencing this sort of discoloration concentrated on that one section. Do you have a fractured door seal that needs to be replaced? You should resolve that before refinishing.