Most importantly, this is historical fiction. Shakespeare tells the stories of English history in a way meant to be entertaining and educational, emphasis on the entertaining. History's great and all, but sometimes it needs a little something-something to really make it compelling to watch—and Shakespeare definitely wanted people to show up and watch.
But Henry VI, Part 1 fits into a few other genres, too. Lots of battles? Check. Swordplay galore? Check. Comrades at arms fighting and dying together? Check. Written to be performed on stage? You betcha. This is definitely a war drama.
The play also has a lot of growing up in it, which places it in the coming of age genre as well. Talbot's son grows up, courageously taking on responsibility in the battle that kills him; Joan of Arc learns to lead and then faces defeat (two necessary components of growing up); and, of course, Henry VI gets, well, older. Jury's still out on the whole growing up bit when it comes to him.