I also equipped the Second Amendment amp, but I didn't have the Hairspray Bomb yet so I'd say just use whatever weapon you have that can kill the quickest, then memorize the order of the challenges (grind kills>melee kills>undergrind kills>bounce kills>blower OD kills>electric barrel kills) so you can already be heading towards the next challenge when you finish one. Completing challenges is the key, here's when I finally pulled it off: