Lee takes you on with the mercenaries once you do a job for one of his men. Mixter lost a book in the old shipwreck off the coast, and getting it back means clearing the beach, diving down to a sunken boat, and pulling the book from a chest near the helm before your breath runs out. Bring 
lockpicks in case the chest is locked, and sort out your underwater breathing first if it is short.
The only real obstacle here is the dive, so get set up before you commit to it.
- Healing potions.
- A decent weapon.
- Lockpicks, in case the chest is locked.
- Better underwater breath if you can get it.
If you cannot hold your breath for long, see 
Homer at the New Camp first. He can train you to stay under longer, which takes the panic out of the dive.
Start with Lee at the New Camp. Once you have talked, he sends you to meet Mixter down at the beach.
Head to the old shipwreck beach, the stretch of coast with the broken ship by the water, and go down toward the sea. Enemies are waiting on the sand, so be ready, though Mixter fights alongside you and helps thin them out.
Put down the enemies near the shore, then talk to Mixter. He explains what he lost in the wreck and what you need to bring back. When he is done, push on through the shipwreck beside you into the next stretch of beach, dealing with or slipping past whatever is there.
Back at the beach, talk to Mixter again, then wade into the water and swim for the sunken boat directly ahead. You are aiming for the upper back of it, around the helm.
If your breath is short, make the trip back to Homer at the New Camp and train it before you dive. The extra seconds make this easy instead of tense.
Swim to the very top and back of the boat, where a chest sits. Open it, picking the lock if you have to, and take everything inside. The one that matters is Mixter's book. Then turn and swim straight back up before your air runs out.
Surface and return to shore. Talk to Mixter, who is glad to have his book back. With that done, travel to the New Camp, find Lee, and tell him you helped Mixter. That wraps up the quest and your place with the mercenaries.