This should be a fairly easy project for you. The first step it to determine, if possible, if the existing paint is an oil or latex based paint. If you have the bucket that the old paint was in, you're in luck. Otherwise, it is hard to tell even though a trained eye may be able to tell. Regardless, using latex will be safe over either oil or latex (oil over latex will wrinkle and peel almost immediately and make a mess).
Since black will be hard to cover with one or two coats of white, I'd use a good blocking primer like Kilz or any good color block. Two quick coats of primer should be a great backer for white over black. Then just roll on your good quality, hard finish floor latex enamel. Probably two coats will do the trick.
Be sure to clean the existing black finish well, and if possible, sand lightly just to rough up the surface a bit. This will help assure a good bond with the primer. Good luck.
Clay Absorbent.
This stuff is available at most auto parts repair stores. Sometimes it is available at big box stores.
It is NOT the same as cat litter although it looks strikingly similar. The resemblance is so close that "kitty litter" is the 'street' name for this product.
However, clay absorbent is far superior in its ability to remove oil from concrete. For example, my father and I moved a broken 2.5 hp motor over a concrete walkway in a plastic tub. At some point the load became unstable and the engine tilted enough to force us to drop the tub. In doing so, we streaked the walkway with oil approximately 2" wide and 6" long.
Since I was away from home and he did not have any clay absorbent on hand, I promptly drove to purchase some and placed the product according to the manufacturers directions. After the second application was swept up, I made a third application and ground the product into the concrete with my shoe and left it to sit until the next rain washed it away (not recommended, but hey the walkway was pristine beforehand and my dad has a low carbon footprint).
To this day, there is a spot on the walkway that is cleaner than the rest. I am not exaggerating in the least when I say that the entire section now needs the ground-in-and-washed-off treatment to help minimize the white spot left behind where there was a dark black oil spot before.
Two applications should do for what you need. Skip the ground in trick unless you promise to recover the waste water ;)
Edit:// Just read where you said spots are old and kitty litter will not work.
Try the clay absorbent. It is about 100% more effective than cat litter on any day. Also, if it does not come up, leave it for a night or two.
Other than that, (using a test spot!!!) dampen the concrete with fresh oil (detergents good...) then place the oil dry over it and leave for at least a day. If all else fails, grind it in!
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For mechanical removal of paint from floors....
If you're talking about drips/splatters/spills that are on the floor from painting the walls/ceiling you can use wire or bristle disks that are meant for removing paint/rust and there's also the Wagner Paint Eater.
If you're talking about the whole floor covered in paint you can rent a floor sander with an abrasive diamond pad or a shot blaster.
All of them are messy and generate a lot of dust. Shot blaster also removes any weak/soft concrete at the surface which could improve your tile adhesion.