Very odd case...any help?

Age: 18
Height: 6'3"
Weight: 152
Fat: Very Little
Muscle: None
Sport: Basketball

I want to be able to dunk the ball. Right now I can barely touch rim. I have a great jumpshot and dribbling skills but I cant get up high enough to finish. Thats really sad for a 6'3 guy.

Goals:

1) To be able to dunk ASAP.
2) I have this protruding sternum. Its solid bone. I think I have a pigeon chest, but then again I have NO pectoral muscle. I can feel the bones under the pecks. I want to basically look normal.

Can anyone suggest some light exercises that take less than 30 minutes per day and do not require weights?

you could look into plyometrics for more explosive power.  Theyre supposed to really help your vertical and dont require heavy weights.  I really really suggest doing squats and jump squats though as if you combined these with plyometrics youd probably see good results.  Im a fraction of an inch under 6 feet and used to not be able to touch rim but now im about an inch away from dunking and its just from squatting basically because ive never done plyos.

as for your chest...just bench...weight training is all i can really tell you.  Your not going to see any kind of significant chest gain doing pushups...

working on explosive phase
vertical: 35
weight: 206

Big 3: 300/440/515

Strength and growth come only through continuous effort and struggle.

Do some jump squats, start with the 45# oly bar and work your way up.  Pretty soon, your body weight will feel a lot light in comparison and you'll jump higher.