Most of this writeup will be focused on the later YJ's (93-95) and the TJs (97-06) as their bushings are a bit a trickier to remove than other Jeeps. Most other Jeeps you will just knock the bushings out with the included removal tool and install the new bushings (do yourself a favor and order a kit with a removal tool). So the later YJ's and all of the TJs are a bit more difficult because the bushing is about half the depth of the hinge (so it doesn't extend to the bottom).
Most of these kits will tell you to use a punch from the bottom, find a ledge and tap upwards. If you do this be very careful. I found it much easier to "e;attack" it from the top. Using a small chisel or screw driver you can get under the lip and work it up. Note here I've only lifted the inner sleeve:
The outer needs removed as well. Again using the same method as before working from the outer lip and lifting:
I had one that was particularly stubborn and I broke it free around all the edges. This also had the benefit of giving me a nice "ledge" to tap on from underneath.
Once you get the old one out, clean everything up. The new bushing should just push in by hand. Delrin can be damaged by beating on it with a hammer, so be careful.
Make sure to clean up your hinge pins. Taking a bit of fine emery cloth or fine sand paper to them is will help. Remember you aren't trying to change the dimensions of them, just smooth them out some. This pin has been cleaned. Notice it isn't shiny, just smooth. That's all it needs to be.