A "Momiji Matsuri" festival is held every year from the end of October to mid November with road stalls and romantic night illumination so it is a popular destination during November for people to see both Mt Fuji and red leaves.
3 day Fuji Hakone pass for 7,400yen and JR offers a 2-day round trip ticket for 5,600yen. We were only going on a day trip so we opted for the JR round trip ticket that includes sitting in a limited express train called Kaiji 101 that you can board within the JR platform at Shinjuku station.
If you are not going up to the fifth station and planning to go around Lake Kawaguchi, then I recommend just queuing up and buying separate tickets that will cost around 2,500yen one way. However if you stay far from Shinjuku, then you might just break-even to get the round trip ticket as it also includes ride within JR Yamanote Line.
You can also take local trains, but that will take around 3 hours for just one-way and costs around 2,000yen. Highway buses are cheaper at 1,750yen and takes 2hours but tickets get sold out pretty fast.
With that, I definitely recommend staying at Lake Kawaguchi for 2-3 days (including going to Gotemba Premium Outlet or Fujikyu Highland amusement park). Oh well, maybe next time!