Is Zero a mutant? In the movie, it seems that he has good skill in using guns but no real power.
Presumably, yes. He was a member of Team X, which was Stryker's black ops team of mutants.
While his powers are never clearly spelled out, they seem to include superhuman accuracy with guns and enhanced agility/strength/reflexes. Since his powers are so subtle, it's easy to be unsure if he's really a mutant or not, but his team affiliation is our best indication.
While he is depicted as an Asian in the movie, David North/Agent Zero is based off of an existing character from the comics by the same name and a different ethnicity and a slightly different power set.
In the comics, North is a confirmed mutant who absorbs and redirects kinetic energy, and has enhanced healing. He goes by the codenames Maverick and Agent Zero at different times. Like in the movies, he was a part of the Weapon X program with Wolverine.
According to Xmenmovies, Zero/David North's powers are related to his skills with guns:
Superhuman Marksmanship - North was an extremely lethal marksman, to the point where he was able quickly draw his gun and shoot Wolverine's cigar without hitting him. Zero is also very skilled with most weapons (usually sidearms), and is very perceptive of the situation around him. For example, during the Nigerian mission, he was able to take out several guards by using his superhuman agility and reflexes in conjunction with his marksmanship. It was also shown when he ran out of ammo, he could reload quickly while still moving.
Superhuman Reflexes - North's reflexes were enhanced to levels unobtainable by normal human beings.
Superhuman Agility - North's agility, balance, and bodily coordination were enhanced to levels that were beyond the natural physical limits of even the finest human athlete.