That brown gunk is grease from cooking. Try a grease cutter and an old toothbrush or even those thin scouring pads--the kind without the sponge and no rough spots that scrape just a smooth abrasive. They will allow you to get in the cracks of the grate. Enamel is pretty tough and doesn't scratch easily.

I use lestoil on mine straight from the bottle and then rinse. Even a dish soap like dawn may work.

As for the handle, wash it too but it may be discolored if it's plastic. If you get the rest clean though a slightly yellow handle may not bother you. You could see if it can be replaced.

I used Windex spray on the one I had in my last house. Also GE. It took the yellow color out of the handle too.

Try Bon Ami powdered cleaner.

It works amazingly well!

Here is a link that might be useful: Bon Ami cleans without scratching

Thank you all for your help. I'll try it all until something works!