How do I dig out a hole for fence posts in a concrete patio?

What you want is something like this:

Get some Red-Heads or other appropriate concrete bolt and fasten it down. Slip the 4x4 in and tighten it up.

Ideally, you should still have some posts properly embedded in the ground to help support the torque caused by someone leaning on the fence, but you can use these in between.

If you have to do this by digging down under the concrete, here's an outline of how I would go about it:

1. Consider renting a motorized earth auger to drill the actual hole. Unless the hole to be dug is very shallow this can be a big labor saver. Otherwise, you will probably need a post-hole digger. Either way the tool to be used will determine how big of a hole in the concrete you need. That said, you definitely will need a good margin around the tool width to be able to work efficiently.

2. Use a diamond grinding disc on a demo saw or even an angle grinder to cut through the concrete. I'd not make this any larger than necessary mainly to save work. Doing this will make for a clean cut on the top surface of the concrete, which is most important. But cutting with a disc like this is fairly easy. Use ear, eye, and respiratory protection. If the concrete is too thick to get through entirely, cut what you can into a grid of ~1" squares and then you can pretty easily hammer those out. The lower portion of the concrete below the cutting depth could be knocked out with a rotary hammer in chisel mode, a cold chisel, sledgehammer, etc.

3. Dig the hole & set the post. (I think setting the post itself is a separate topic entirely.)

4. Once the post is in place, neatly fill around it with concrete to restore the patio surface.