There are 5 challenges to complete for the Explosive Rifle. After you've finished all 5 challenges, the achievement will unlock. The challenges are as follows;
1. Purchase the Explosive Rifle
The Explosive Rifle can be bought from the Gunsmith in Blackwater. It's price will be one of three options;
The price will depend on your fame. (Honor?) The higher your fame is, the lower the price of the gun is. Being negative in fame will cause the rifle to be $15,000. Somewhere around Neutral will cause the rifle to cost $10,000. Be extremely positive, and the rifle will cost $5,000. If you happen to have very low fame, and low money, try and bring your Fame up. You can do this by wearing the Elegant suit, cheating at Poker and shooting the gun out of your opponent's hand during the duel after being caught. If you play at MacFarlane's Ranch, the buy-in is only $10. Repeat this over and over until you are at high enough Fame where you can buy the rifle at $5,000.
2. Kill 5 Enemies while they are in cover
After you've purchased the Explosive Rifle, you'll need to kill 5 enemies while they are in cover. Traveling to the nearest hideout is the easiest way to find enemies. When an enemy is in cover, they'll be out of sight, and hiding behind something. This is when you must take your shot. Repeat 5 times, and the challenge should be complete.
3. Kill 3 Enemies With One Shot
Killing 3 enemies with one shot may seem like an improbable task, but it is much easier than it seems. Hog-tying 3 enemies and placing them within a spot where they're all next to each other, will do the trick. Shooting your enemies from a higher point may make the job easier, but that is down to personal preference.
4. Kill Two Wild Animals
Killing any type of wild animal will work. Vultures will spawn after you've completed a hideout, bait can attract different types of animals, and finding random spawning animals all work well. Be sure to use Dead Eye when you're trying to shoot a hard target.
5. Kill an Enemy While Wearing the US Army Outfit
First, you'll need to have unlocked the US Army Outfit. This outfit is tied to the Redeemed achievement. For information on how to unlock it, check this thread. You can change your outfits at your nearest Bed. Change into the outfit, and kill an enemy.