I am thinking about taking up the tile in my kitchen and staining the concrete underneath it, but I don't know if it will be possible to get up all the stains left on the concrete from the tile. I do not want to take up the tile if there is too much of a possibility that the tile stains won't come up or that the concrete stain won't cover it up.

It will take some work to get the concrete is good shape for staining. I have tile on concrete and the morter was really tough to get up as the concrete was not sealed before it was tiled.

I am unsure about your tile, ceramic or is it vinyl? If you wanted to do this job you would need to have the cement floor sanded with a floor grinder. Do not expect to pop the tiles and stain the old floor, its not that simple.