There are buses to both LaPush and Neah Bay from Forks. Also, one to Kalaloch and Lake Quinalt. Cheap transportation. The transit center is on hwy 101 and you can catch one there for all those destinations.
I wouldn't recommend hitchhiking. There are two prisons in the area, one being an honor camp, meaning there is not much preventing an inmate from breaking out, but his honor. There are escapees quite frequently. Some years ago, when the police were looking for an escapee, all the hitchhikers between Port Angeles and Aberdeen were stopped. Each and every one of them, about 15+ men, had a warrant out for his arrest. Scary. A young man hitchhiking from Port Angeles was picked up by a recently released prisoner on his way to Forks from Monroe. The young hitchhiker ended up stabbing the man who picked him up to death to save his own life when the man tried to rape him.
I don't usually recommend hichhiking, but I did it all over the penninsula - incredibly safe and I'm a single girl. Buses are useful but infrequent.
Be sure to see Ruby beach - one of the most beautiful places I've seen in my life.