Wouldnt bother with dvd cleaners personally if they even work.. as that won't clean the insides or anything else so any problems may still occur
I'd start off with googleing for a guide on opening it.
You basically flip the ps2 upside down, and take all the little rubber feet off ( using a screwdriver to prong them up if needed )
then you flip it back over, slowly so nothing falls out!

and then holding the front of the top casing slowly open it being sure not to rip the cable that its joined onto
Then you need to remove the CD/DVD cover which can be tricky when you are first doing it ( remembering to look for the tiny little screws) - this has a yellow label on it and has a texture to it you can't miss it

best to use a youtube vid of someone opening it up to get the general idea
Then you are at the laser, you can use a damp cottonbud to give it a clean in a circular motion and then a dry cottonbud doing the same
I'd highly recommend having a look at youtube for ps2 repair vids and opening etc as it will be more indepth and you will be able to visually see what you need to do
While you are cleaning the laser you may aswell oil all the cogs etc and dust everything
I did mine about 3 years ago now when it started freezing on games and it still works perfect to this day

i felt like a true repairman!
Also don't forget you'll have to take of the warranty sticker on the back of the PS2 to open it, not that this matters thesedays as i don't think anyones would still be in warranty if they even still do it