Nogy's video above is pretty good for showing the cotton going in.
3 tips:
1 - Get a point at one end of the cotton to get it started through the coil. That pointy tip will get trimmed off, so it doesn't matter if you wet and twist it...
2 - grab that tip and pull the rest through. Regular tweezers work OK for this
3 - wick should then slide back and forth without stressing the coil, but brushing it all around - if it's close to that, you have about the right thickness of wick - and you'll get the hang of how much to cut to start off pretty soon...