What I normally do, is completely wind the spring so it's small enough to fit. Then I take needle nose and pinch it somewhere so it doesn't unwind. Keep in mind, winding the sucker makes your finger tired...
With it in the needle nose, I place the outer hook in the crevase it needs to be in as I hold the spring inside the cup where it needs to be. Slowly release pressure on the pliers and the spring should pretty much end up where it needs to. You may want to keep a thumb on the coil as it releases tension and unwinds a little in the cup. It's easier to wind it tighter than it needs to be, then let it unwind in the cup.
Once it's unwound in the cup and you don't need to hold onto it anymore, just wind up most of the string on the spool in the right direction and try to place the spool in the cup while eyeballing where the notch on the spool fits to the spring.
Then watch the movie a couple times toward the end to see how to take out the slack. Actually, don't watch the end of the movie to take out the slack, that was just a PITA!!! There's an easier way, if you can't figure it out, follow the vid. To hard to explain.
Damn... it is hard to explain. I've done it like 20 times, but never wrote it down... Oh well, hope you get the jist of it.