OK, here's my method. First, get a solder sucker, there are a couple types. The sucker-only types work, but you have to be very fast to heat the joint, then replace soldering iron with sucker and trip the sucker before the joint cools. Or, there are types that have the heating element with a hollow solder tip/nozzle, that gets rid of the quick switch problem.
So, use the solder sucker to remove the old solder, it might be lead-free, and need a LOT of heat to melt. Then, replace the old solder with lead-based (or low-temp solder, if you have it). Replacing the high-temp lead-free solder with even tin/lead solder makes a HUGE difference.
Now, remove the component. If a big radial capacitor, you may have to alternately heat one lead and then the other, and rock it off the board.
Once the component is removed, then you can clear the hole. If using the sucker-only device, apply the soldering iron to whichever side of the board has more copper on the pad, let it heat for 10 seconds or so, then apply the sucker to the other side and trip the sucker while still holding the soldering iron to the other side.
If using a combined heater/sucker, then you can put a soldering iron on the other side, thus heating from BOTH sides of the board. This works amazingly well on heavy multilayer boards like computer motherboards. Make sure the through hole in the board is FULL of solder before doing this, the solder will help the entire hole be molten, so it gets completely sucked out.
I've not had ANY luck with desoldering braid on multilayer boards through holes.
Jon