How do I tile around my fireplace?

You need to remove the slate

Yes, the slate would need to be removed. Patch any damage then tile.

You need to remove what is there, then prep the walls then install new tile. See if these help you.

https://www.thehandymansdaughter.com/install-fireplace-tile/

https://www.hgtv.com/design/decorating/design-101/how-to-tile-a-fireplace

https://www.thisoldhouse.com/how-to/how-to-tile-fireplace-surround

Have you tried using mastic to hold the tile in place? That's what's used to hold tile on the walls for like showers.

If you really don’t want to remove the slate, you could get a drill bit for slate and then screw on some thin plywood over the slate and tile that. As long as the wood didn’t make the tile hang over the edges. Or you could get that peel and stick fake, textured tile and use a strong adhesive. At least that way if you wanted to change it back you could.

Here is a different approach ( it is also virtually FREE) : leave the slate and draw the eye away from it so it is just the background. Make your mantel the focal point with seasonal items in your decor style. Put a lovely picture above the mantel or on either side of your mantel at eye level. You could also add a tall pot or crock ( whatever as long as it is not flammable) at your hearth area for a bit of color. NOTE: I have the exact same slate with white mantel and gas logs. Best wishes!

https://www.youtube.com/results?search_query=how+to+tile+around+fireplace

It's a gas insert. The slate is glued to wall around it. Easily removed. Pry it off with a stiff putty knife. May be some damage to the wall. No shrink drywall patching compound to repair any damage.

You don't have to remove the slate! There are tons of construction adhesives that will practically stick water to water. (joke... for those of you that are not so quick). Go to Home Depot and ask them to show you choices. I am using one that bonds immediately but you can still slide around to make adjustments. It is by Loctite and is called Power Grab. Love it.

Paint it . Or stencil a design on it . Slate takes paint very well.