About forty years ago the previous owner of my house had the floor of the garage portion of the barn covered with asphalt. It's still in good shape, but I'd like a smoother surface, like concrete. Or maybe tile.
Would a thin layer of concrete just break up?
Is there an easy way to lay tile on asphalt?
I would pull the asphalt up - a thin layer of concrete on top of asphalt would be prone to cracking and may break up easy. You want a good consistent thickness of concrete on top of a properly compacted granular base.
Some advice you receive on the internet can be worth exactly what you paid for it - be careful.