I don’t have a LCP but I use a drop of CLP on a Q-Tip to clean the firing pun chamber on all of my semiautomatics. I then use dry Q-Tips to insure there is no residue CLP left in the channel. I also don’t clean the channel until I have fired about 600 or 700 rounds.
The Q-Tips I use are the ones with a wooden stick not the plastic ones.
There are good disassembly videos on YouTube. From what I viewed, when you field strip the slide from the frame there is a pin on top of the slide that must be driven out to remove the firing pin and spring, then you should be able to access the channel. Remember I do not own a LCP. Good luck.