Pinpointing the "first" violent video game is difficult because definitions of violence have evolved, and early gaming history involves many experimental projects. However, three titles are frequently cited as pivotal early examples:
"Violence" is relative. Early games used vector graphics or text; modern games depict realistic gore. The evolution runs from abstract collision (Spacewar!) to controversial depiction (Death Race) to the complex narratives of today.
Ultimately, the answer depends on your threshold: first combat mechanic (Spacewar!), first controversial depiction (Death Race), or first interactive electronic game (Tennis for Two).