Fly from Santiago SCL to Puerto Montt PMC on any national airline:
Sky Airline | Jet Smart | LatAm

From Puerto Montt fly to Chaitén on a charter company:
Pewen | Aerocord

There are also buses that take you from Puerto Montt directly to Futaleufu:
Buses in Chile

From Puerto Montt to Chaiten by ferry, 10 hours. Must be reserved with anticipation. If you’re travelling by car, you must have the license plates and the name and passport number of each passenger:
Naviera Austral | Transportes Austral

If you are a pedestrian leaving from Pto. Montt, Kemel bus is a good option:
Kemel Bus

It’s a 4-hour ride from Chaiten to Futaleufu.

Arrive in San Carlos de Bariloche. Continue South to Esquel by car or bus:
Buses in Argentina

Continue on to Trevelin and then cross the border into Chile to the town of Futaleufu.

There are also buses that take you from Puerto Montt directly to Futaleufu.

Esquel Bus Station