How do I paint my kitchen ceiling?

https://homeguides.sfgate.com/clean-kitchen-ceiling-covered-smoke-residue-93322.html

https://homeguides.sfgate.com/clean-grease-kitchen-ceiling-92879.html

First, you'll need to clean the grease and smoke off as much as you can with a degreaser like TSP (trisodium phosphate) found in the paint section. Then prime over it using a stain blocking primer, like Kilz. After that, you can use ceiling paint. They make ceiling paints with primer built in, but for heavily stained ceilings, I'd go the extra step and use the Kilz first so you know the stain won't bleed through the paint later on.

It depends on whether your ceiling is smooth or stippled. If smooth, wash it with a degreasing cleanser, rinse well, let dry completely, then prime with a stain blocking primer and repaint. If your ceiling is stippled, you can't wash it or the stippling will come off. You will have to very carefully prime the area with a primer made for blocking grease. I would use an oil based primer because a water based primer could cause the stippling to come off. (I have painted a whole house with stippled ceilings.) Make sure that you only go over the stippling with one motion. Even if the primer doesn't cover completely resist the urge to go over the area again. The stippling will come off easily when wet. After the primer is dry, do a second coat. Then paint as usual.

getting it CLEAN is your first step ... can't paint over the smoke/grease!

https://www.thriftyfun.com/Repairing-After-a-Fire.html or

https://www.angieslist.com/articles/5-steps-clean-after-small-kitchen-fire-your-home.htm or --

https://homeguides.sfgate.com/clean-kitchen-ceiling-covered-smoke-residue-93322.html

Clean it first with a decent sponge roller and some soap, bleach and Degreaser, then rinse it well.

After Cleaning it you may find you less need to paint and will save $$$$ By Cleansing regularly rather than painting.

However, if it is more than a Coloration problem then you do need paint. I recommend using Kilz Brand Paint in the Kitchen. It stands up better to Grease and also: Degreasers that you need to use at least once each 6 months on a white kitchen ceiling.

I'm a painting contractor. Depending on how much smoke smell, you might want to use a odor blocking primer or paint. Sherwin Williams carries some.