Although it would be easier to complete this mission once all the weapons and their upgrades have been acquired, one can also employ an alternate strategy of pushing the enemies off the platform leading to an environmental kill. Dante can use Arbiter's Flush and Aerial Flush to push off the Lesser Stygians and Eryx's Snake Eye and Uppercut to push off the Elite Stygians. If Dante has already acquired Aquila, he can also use Buy In, activate Tornado and move into lava, pushing the enemies to environmental kills.
This one becomes super easy as soon as you realize that the counter will be reset every time you drop down in the lava. (May need confirmation. Discovered on PC version using the Stinger attack.)