Bodybuilding with dumbbells

I'm about 5'5, 130 lbs. and 20 yrs. old. I hope this doesn't sound redundant. I lost a good amount of weight about a year ago and I've been very successful in keeping it off. My problem is that now I think I'm too skinny. Sometimes I feel like a twig even though I'm at my perfect weight. I'm even a little flabby in my chest, arms, and legs. That concerns me from time to time.

Needless to say, I'm new to bodybuilding but I've been doing a lot of research so I think I know some of the basics (nutrition, reps, etc.) and what they mean. My main goal is to gain around 20 to 30 lbs. of muscle. I know it'll take a lot of time and hard work but I know that I can do it. I don't think it's impossible either.

I really don't like the gym atmosphere so I'd rather work on this on my own and in my own space without distractions. I worked on my weight loss on my own too (treadmill, park, body-weight exercises) so motivation wouldn't be a problem. I've been looking at some Iron-master dumbbells. They go from 5 to 75 lbs. Would those be a good investment? And would my goal be achievable with that equipment?

I'd appreciate any suggestions. Thanks a lot!

To be quite honest, a gym atmosphere is essential for remaining motivated.  Doing workouts with only dumbbells is possible, but deadlifting and squatting, which are essential exercises, are not as effective using dumbbells.
 
If I worked out at home, I would lose motivation.  Difficulty in getting the right kind of equipment (cable crossover machines for example), and a lack of variety of plates, bars, and dumbbells would really hold someone back.
 
YMCA is a great budget gym.  I pay $35 a month.

Squat 385, Bench 235, Deadlift 450 @ 181.

Motivation comes from within. Some find it better in the gym, some like the privacy of thier own home.

your goal is certainly attainable, however if you want to build significant mass, you may want to opt to go higher than 75 lbs, by purchasing a barbell as well (since freeweights are your forte) so you can stack a heavier plateload as your stength and body adapts to your routine. Dont forget the longer and more frequently you lift, the more strength you will acheive, and you will want to increase your plateloads to bypass your plateaus.