Gloves are the primary way to perform elemental attacks toward this achievement. When examining a glove, below the Strength and Defense stats will potentially be one of the elements you'll need to complete this achievement with a random percentage after it. For example, a glove stating "Fire: 12%" has a fire elemental attack with a 12% chance of the enemy bursting into flames.
The glove elements are:
Poison: enemies throw up when they die;
Fire/dark fire: corpse turns black and burns;
Acid: skin melts off with a bubbling sound effect ("melt" requirement);
Thunder: skin is burned off with an electric buzzing sound ("shock" requirement);
Frost: enemy turns blue and makes a shatter sound when punched ("shatter" requirement).
The enemy must die by the obvious elemental damage as outlined above. Simply having the glove equipped and then killing an enemy with it does not contribute an elemental kill toward this achievement.
As for the "explode" requirement, there are no gloves with that effect but the grenades you'll find littered about the game will help you out there. If you need a specific location for some grenades, Art Street has tons of them on the movie theater screen (where the blockade of bombed-out cars is.)
Frost seems to be the rarest (at least in my experience) and is probably the only element you're missing if you routinely equip stronger gloves over the course of the campaign. Fortunately, Kelly's Evil Eye is a frost-based attack and counts toward this element (which is the very first tattoo you come across right at the beginning of the game). It's also been reported that the Lich version of Lester begins the game with an icy riff anar-chi attack, which counts the same as Kelly's evil eye toward a frost kill. Conversely, experiments with all elemental relics (acid thrown objects, poison grabs, electric throws, etc.) seem to indicate that none of them will help you out with this achievement.
These also do not need to be accomplished all in the same playthrough or all with the same character. If you finished a playthrough and still do not have this achievement, chances are good that the first enemy you shatter with Kelly's Evil Eye will net you this achievement.