****This solution is for the DLC only. If you have purchased or rent the Undead Nightmare Disc, please see the solution provided by either The Recluse or osubluejacket.****
There are 5 ranks for the Explosive Rifle in single player.
Rank 1 - Obtain Explosive Rifle from gunsmith in Blackwater, be sure to buy max ammo available at the gunsmith for the rifle, you may need it. Cost of the rifle depends on your honor. Lower $15,000, middle $10,000, high $5,000 as confirmed by Rusty Nail zh. If you don't have high honor and don't want to spend 10k or more for the rifle you may want to start duels with people and just disarm them to move your honor up.
Rank 2 - Use the Explosive Rifle to kill 5 people in cover - I went to the Twin Rocks gang hideout, didn't actually get all 5 in the first time, but, finished it after setting up camp to save to respawn the hideout. You could probably go to other hideouts that have more enemies to complete this faster than I did.
Rank 3 - Kill 3 Criminals in 1 shot using the Explosive Rifle. I used what GreenBeretBrown suggested, went to Fort Mercer, hogtied 4 criminals (I did 4 just in case) and shot into the pile.
Rank 4 - Kill 2 animals with one shot with the Explosive Rifle. After I cleared Fort Mercer and the vultures are flying overhead, used dead eye to target one of the vultures when it flew next to a 2nd Vulture. I tried it without dead eye, but, I couldn't get it to auto aim on a vulture. Others have mentioned using bait to do this part and shooting the ground between 2 animals once the bait lures them in.
Rank 5 - Obtain or change into the US Army Uniform and kill an enemy.
If you have access to the US Army uniform already, you may want to change into it before starting on these challenges. Went to a safe house and changed into the uniform and there was an active bounty on the bounty board, accepted it and killed the first enemy using the explosive rifle. By completing this challenge you get a 50% discount on explosive rifle ammo and the achievement.