What is the console command(s) to modify the lockpicking skill?

It's not advanceskill, it's AdvSkill

See here: Skyrim Wiki

Note that advskill lockpicking 1 will raise both your lockpicking skill and player level, while player.setav skill level will only raise the skill level without leveling up your player character.

Alternative to AdvSkill which was mentioned in the answer of Asunez, there also is incPCS. This is fairly safe since you only modify the skill itself and you can't modify something "accidentally".

incPCS lockpicking

This will increase your lockpicking skill by 1. The difference to AdvSkill is, that it doesn't matter how much experience is missing. It will simply fill up the bar. So you enter this once and then you simply spam it over and over again. Or you can copy it into a .bat file and execute it ingame via console. This is - in my opinion - the easiest way to gain fast skill levels if you really want to use the console for that.