It is incredibly difficult to gain muscle while losing fat. The two goals just have different requirements. You can do lean gains and eat around maintenance, lift heavy and see a bit of both, but the results will be slow.
In order to build muscle, the body needs fuel, so it requires eating at a caloric surplus. Usually one would go 200-250 calories over maintenance, dial back the cardio and lift, lift, lift. You would also gain fat, but having a smaller surplus would help to, hopefully, miinmize that.
Fat loss requires a caloric deficit.
I'd suggest that you lift heavy and hard while cutting for awhile and see how your feel. Then you could do a "bulk". I would up your protein to at least 1 gram per lean body weight. My husband lives to build strength, and he tries to eat 1 gram of protein per pound, sometimes over that, aiming for per pound he'd like to weigh. Yeah, he eats LOTS of protein. Boydbuilding.com and muscleandstrength.com have some good guides, BB.com has good forums, too. There are some jackasses on there, but some very useful info. You might look into New Rules of Lifting or Starting Strength. My husband also likes testosteronenation.com.
If you are relatively new to weight lifting, I'd start with a solid full body workout, squats, bench and deadlifts for sure. Then, depending on you, pull downs, pull ups, push ups, overhead press, shrugs, calf raises, bent over rows also.