One option is to use moves.
- Double-Edge has recoil damage, but Jigglypuff learns it at level 49.
- Perish Song would make Jigglypuff faint in three turns, as well as the opponent, but it's an Egg move.
- Substitute will use 25% of Jigglypuff's HP to make a Substitute. You can find it as a TM on the second floor of the Old Chateau, or buy it in the Veilstone Game Corner for 2000 coins.
There's also held items.
- Life Orb will take 10% of Jigglypuff's HP when he attacks a Pokémon with a damage-dealing move, but it's only obtained at Stark Mountain.
- Black Sludge will take 1/8 of Jigglypuff's HP every turn, as he's not a Poison-type. It can be stolen from wild Croagunk and Toxicroak, though the chance is only 5%.
- Flame Orb burns Jigglypuff and can be bought at the Battle Park in Diamond/Pearl or the Battle Frontier in Platinum for 16 Battle Points.
- Toxic Orb badly poisons Jigglypuff and can be bought at the Battle Park in Diamond/Pearl or the Battle Frontier in Platinum for 16 Battle Points.
- Sticky Barb damages Jigglypuff by 12.5% each turn, but transfers to the opponent if it hits him with a contact move. You can buy it at the Veilstone Department Store or steal it from wild Cacnea and Cacturne, though, like Black Sludge, the chance is only 5%.
The most simple method is letting the Pokémon you're facing attack Jigglypuff until he faints. This is quickest against powerful Trainers. You could also enter a Double Battle and have your Pokémon attack Jigglypuff. If he's at a high level, Double-Edge plus a held item like Black Sludge can help him along.
Tell me if I'm missing something or didn't show you what exactly you're looking for.
Source: Bulbapedia's item, Pokémon, location, and move pages (I checked more Bulbapedia pages than I linked here of course but these are a few of them)