This is a good way to see some Hakone highlights without being herded through the typical route. It is a popular walk, so while you won't find solitude, at least it isn't on the widest part of the beaten path... A lot of it goes through wooded areas, so it's a good way to spend one of those days when the elusive Fuji-san is hiding in the clouds.
Take the Odakyu train line from Toyko to Odawara. From Odawara, take Hakone Tozan Railway (300Y) to Hakone Yumoto Station. From there, take a bus to Sekisho-ato Iriguchi bus stop.
The hike ends at Hatajuku or Sukumogawa bus stop. From there you can take another bus back to Hakone Yumoto Station.
The hike is 5 to 11 km. It takes 2 to 4 hours. There are place along the way where you can jump off the trail to bus stops.
Start at the Old check-point, Hakone-machi. Admission is 300Y, if you want to go in. You can also detour off the path to see the Hakone Detached Palace Gardens (Y?). Walk through the cedar avenue. At Moto-Hakone detour off the trail ~15 minutes to see the tori gate along the Lake Ashi shore at Hakone Shrine. Back on the trail at Moto-Hakone, continue walking for about 45 minutes. Where the trail meets back up with the road, stop at the teahouse Amazake-Chaya. Here you can get a cup of sweet rice alcohol, amazake, for 400Y. About 35 more minutes up the path you arrive at the Wooden Handicraft Center (free). From here you walk along a stone paved road which is the Old Tokaido. There are bus stops at Hatajuku ~5km, Sukumogawa ~6.5 km total, or you can walk all the way to Hakone Yumoto ~11km total. The longer walk goes by numerous temples as well as the Tenzan Notemburo Onsen, 2 km from the station (1100Y).