Limited express bus
Shinjuku to Kawaguchiko station. Direct, one seat, runs every 30-60 min. Cheapest comfortable option.

Distance
About 100 km in a straight line. Two hours door to door if you pick the right train. Here is every option, ranked.
Shinjuku to Kawaguchiko station. Direct, one seat, runs every 30-60 min. Cheapest comfortable option.
JR Chuo line to Otsuki, change to Fujikyu line. Covered partly by the JR Pass.
Tokyo to Mishima on the Tokaido Shinkansen, then bus to Lake Kawaguchiko or Gotemba. Fast, fully covered by JR Pass to Mishima.
Chuo expressway. Beware weekend traffic out of Tokyo from 7-10 AM.
Romancecar from Shinjuku to Hakone-Yumoto. Closest Fuji-view base from Tokyo.
Tokyo Station to Fuji summit is approximately 100 km west-southwest.
Tokyo to Lake Kawaguchiko is about 120 km by expressway.
Around 90 km. Hakone is the closest Fuji-view region to Tokyo.
2 h 30 by Shinjuku-Subaru line bus, the trailhead for the Yoshida climbing trail.
If you also plan to visit Kyoto, Osaka or Hiroshima the same trip, a 7-day JR Pass usually pays for itself. Order in advance from getjrpass.com so you have it on arrival.