Be careful not to increase the salinity too quickly. If he is in a 10gal tank, then you can change out 5gal everyday, or every other day as mentioned. Start adding a teaspoon of salt to the new water when you do a water change and make sure it is completely dissolved before putting it into the tank. Do this for a few weeks and then start adding a second teaspoon to the new water for a few more weeks, and so on and so forth. The process will tank months but you eventually want to end up with water that has a SG of 1.010-1.015 (oh yeah, you will need a hydrometer to measure the SG/salinity). I believe that you will end up in the range of 1 cup per every 5 gallons to get a SG of about 1.010. This should take care of the brackish needs of the dragon/violet goby. Just make sure you use marine salt (the stuff used to make saltwater for saltwater fish, not just regular aquarium salt).
Ok, now to discuss the health of this guy. He looks pretty thin. It is not uncommon for them too come pretty thin from the store, especially if it was WalMart. However, it shoul only take a week or two for them to start to thicken up. They are filter feeders and will "inhale" their food. I used to feed mine brine shrimp pellets and they would just sit with them and gulp them and spit them out until the pellet broke apart and they would eat the tiny pieces as they broke off. I would strongly recommend you try some brine shrimp pellets. Also, these fish don't float. The swim around the bottom but they sink. I don't believe they even have a regular swim bladder but instead they have a modified swim bladder that allows them to "breath" air. This is why the fish is gulping at the surface. It is also a sign of poor water quality. In a healthy aquarium, they will stay on the bottom at all times. Even when they swim around, it will be on the bottom only. As mentioned, the tank needs much more than an hour to cycle. It sounds like your tank is still cycling and the fish is letting you know by gulping air at the surface. These guys are very sensitive to ammonia and nitrites.
Now, for tank size, as some have already mentioned, these guys get way too big for a 10gal aquarium. They regularly (and rather quickly) can reach 24" and be as thick as your average wrist size. A 75gal is the minimum to keep them but a 125 would be best, or bigger if you want.
I know that this is a lot to take in but these are not your average fish. When kept under the right conditions, they are very hardy and fun fish to keep. Not really anything more "dragon-like" that I know of. If you don't meet their requirements, then they will suffer and eventually pass away. You are seeing the beginning stages of this with yours but it's not too late yet.
A few questions that can help better guide you:
What are you feeding it?
Do you regularly see it eat?
What type of dechlorinator are you using when doing your water changes?
When can you get a larger tank for it?
Are there any other tank mates in the tank with it?