Slaw, working with AN wire is a delicate process, but with patience and care you can do it. Nandric's suggestion of anchoring the clips (one at a time; of course) on an old cartridge is a good one; you don't want solder "creeping" into the clip's sleeve. This can happen even if you hold or secure the clip vertically. A toothpick stuck in a large ball of modeling clay (or similar) also works well. The advantage here is being able to adjust the angle of the tooth pick/clip easily; horizontal is best.
-With the clip secured to the toothpick or cartridge, tin the clip's tab over it's entire length. Don't use too much solder; you don't need much.
-Now comes the hardest part; tinning the wire itself. After a lot of trial and error, I have found that the best technique is to melt a bead of solder unto the the iron's tip and with your other hand run the end of the wire through the bead of melted solder until you can see that the wire is coated with solder. Don't hold the wire in the solder bead too long or the wire itself will melt. Run it through slowly; it may take three or four times. Once the wife's insulation has melted the wire will be coated with solder; not before. If you try touching the wire with the tip of the iron you will probably melt the wire.
-Now, solder the tinned end unto the prepared clip's tab. I like to make sure that a little bit of solder (remember, don't use too much) runs unto the wire beyond the tinned section. This will provide some strain relief, as the insulated portion will be less susceptible to fracturing.
-Slip the heat shrink tubing over the wire unto the clip, making sure that it slips on past the tab and unto at least part of the sleeve for additional integrity and strain relief. Don't try to slip it on over the clip first.
-Unless the heat shrink tube fits very loosely (unlikely) I prefer to not heat-shrink it. This makes repairs much easier, by allowing you to carefully slip it off of the clip and reusing it.
Good luck, and let us know how you like the AN wire; it transformed the sound of my analog setup.