Cleaing Above the Stove Microwave

You need a strong degreaser. I was looking at mine the other day and thinking it looks gross.
After cleaning so, so many very greasy college apartments, I learned that undiluted Pine-Sol is amazing on that stuff. I would take the knobs off the stove and soak them in it, and years of build-up would come off really easily with a toothbrush. You can apply it to the vents with the toothbrush and it ought not get too messy. Just get it wet with it. Then wipe it off with a rag.
I haven't tried it yet, but that's what I would/will try first.

Here's something else I learned that's related. You know the filters under the microwave (or under the vent hood) that get all coated with grease? If you put it in a clear plastic bag (like a Zip-loc, if it'll fit. Mine didn't, so I found something else) and pour in ammonia, then seal it and set it out in the sun, it loosens all the grease. Then you take the garden hose to it and it all comes off, shiny as new. I did it to my 1971 vent hood. It had been a rental, so I doubt anyone had ever cleaned it---ever! It was really really bad. It came completely clean.

Simple Green is awesome. It is the easiest way I've found to clean the greasy dirt build up in and on my vent... and it isn't expensive. I get mine at Ace Hardware. Just spray on and wipe of with papertowels...hooray!

I use rubbing alcohol( for microwave vents over stove )on paper towels and it takes the built up grease right off. Alcohol works wonders and will evaporate.
Now for cleaning the vents underneath the microwave which get quite a bit of grease build up, I soak in very hot water and Dawn liquid, then later on, squeeze rubbing alcohol over screens and use a brush to clean mesh. Ammonia will cause aluminum products to turn very dark. It should only be used on stainless, but not aluminum.
After brushing with alcohol, then run in Dishwasher to rinse and sanitize the removable screen vents.
Be sure not to use alcohol around flames or heat!

Like everyone said, degreaser. I used Grease Lightning - I just sprayed it on, and put a newspaper on the stovetop and let it sit. Then I wiped it off. I like to clean after with something like windex or another non-oil based cleaner, because oil based cleaners leave just enough residue for dirt to stick to. But that's a tiny detail.

Ok, I do not know if this will work but have you tried magic erasers? Also before you try to clean off the grease boil a couple pots of water so the steam warms up the grease so it is easier to clean.