The Vengance Launcher is obtained in Chapter 3 and the Power Axe is obtained at the end of Chapter 5. You must complete an entire chapter firing only the Vengeance Launcher and swinging only the Power Axe.
Note: Throwing grenades will void this achievement.
An easy method for this achievement is to again load the final checkpoint on Chapter 6. You simply have to finish the mission using only the Power Axe and the Vengeance Launcher. If you load this checkpoint make sure you grab the needed gear from the weapons cache before you run into any bad guys.