Hakone (+ Lake Ashi) is easier to reach than Lake Kawaguchi and if anything less "spoiled". If you're just interested in a short trip from Tokyo to see Fuji and a lake, I would suggest Hakone.

The Fuji Five Lakes region (Yamanaka, Kawaguchi, Sai, Shoji, Motosu) is a nice area to tour if you want to see the contrasts between the big (Yamanaka) and the small (Shoji) the very touristy (Kawaguchi) and the unspoiled (Motosu). But it's a longer trip and it requires a car to do it with ease.