I was in the same position as you are in now. I shot up to 6 foot 2 and I was very slim.
Key is to eat huge amounts. Count your calories.
Also, 15lb dumbells won't be enough. I tried working out at home with 35lb dumbells and they are not enough.
You need to be doing large compounds lifts with heavy weights. For muscle growth you want 3 sets of each exercise at around 6-12 reps. You should lift every other day with a two day break every week. That's 3 days a week or maybe four if you want to. But NEVER two days of heavy lifting on the same muscle groups in a row.
You need to do compound exercises (exercieses that incorporate many muscles). These release testosterone and are essential for muscle growth.
Do NOT do what tentomruler says (no offense). He does not work the legs and his workouts are much too split up for a skinny guy.
You need a full body workout (fbw).
Get a gym membership or plates and a bar because you NEED heavy weights. Gym will be better because right now you are weak and machines are easier.
You need to do the following exercises in one session - it should take under an hour.
Leg press/ squat/deadlift
Bench press
Lat pulldows or bent over row
Shoulder press
Romanian deadlift/ hamstring curl
These work the major muscle groups and will get your muscles growing. You should also be able to increase the weights you lift as you get better at lifting.
I know you don't want to, but it is essential that you work the legs. The key one is the squat/ leg press.
After these five you can then do some isolation e.g. arms and core
You really are going to have to eat ridiculous amounts. I recommend drinking milk. It has large amounts of protein (8g per full glass) and large amounts of calories (125 per full glass). Eat meat, cheese and other foods that are high in fat and protein. Don't worry, if you are that thin, you will have a very quick metabolism so you shouldn't get fat, if you do start gaining large amounts of weight (probably a good thing) then eat less or just lay off the high calorie foods.
Also, get enough sleep so that your muscles can recover and grow and also remember to stretch after lifting so that you don't rip something and so that you won't be sore when you next lift.
It will suck but within six months and definitely by next summer (I know, it sucks that its over), you will look completely different. You could put on 20lbs if you work hard and will actually look very musclar.
Remember: Heavy lifting + Heavy eating + Enough sleep = muscle growth
Good luck and welcome to the forum.