Cleaning a dirty clicky keyboard is a lot of work. It pays to take lots of photos of the keyboard before tearing it down, so you can remember where the keys go.
The best way to remove the keys is to lever them from opposite sides at the same time. Start with the letters and numbers, leave the bigger keys till last.
Notice that the bigger keys have metal retainers that have to be carefully removed from the plastic. The space bar also has a weak spring.
Also, be careful with scrubbing the keys in solvents or harsh detergents, the keyboard labels may wash off.
Putting it back together is twice as fun, hook the keyboard up and run a program like notepad to see if the right key is in the right spot.