Uhh... I have a bad feeling about this.
The Tonberry starts at distance from the party and takes a step forward each turn. When any move is used against it, including attacks as well as non-damaging moves and healing, it will counter using Karma, dealing damage based on the number of enemies the target has killed. After five steps it is close enough, and will attack a party member with his knife.
It's best to kill the Tonberry before it gets close enough to attack. The easiest way to defeat Tonberry is to have Wakka use Sleep Buster. Asleep Tonberries will not counter-attack, and will be left vulnerable to Black Magic spells. Physical attacks wake the Tonberry up causing it to retaliate with Karma.
Another method is to simply hit with the strongest damaging attacks such as Flare, Holy, or the basic "ga" spells. Physical attacks with a strength of at least 40 work too, making Auron a favorable candidate. This method can consume many Phoenix Downs to keep the party alive depending on how much damage Karma does to each member. If Yuna has learned Black Magic, she may be an ideal candidate as she will likely have killed fewer enemies than others due to her aeons having individual kill stats, making Tonberry's Karma deal possibly negligible damage to her.
Another strategy is to use Tidus as support, cast Haste or Hastega on the party with the priority order being Tidus → damage dealer → Yuna. Once everyone has Haste applied Tidus should buff the party's damage dealer with Cheer. Tidus could also play as a damage dealer, but is usually better off as support, as each time the damage dealer is KO'd and revived their turn is delayed. Yuna should cast Life when damage dealer falls. If the damage dealer survives Karma Cura or Curaga can be used to keep them in good health. When not healing Yuna can defend. If Auron is used as damage dealer, he should physically attack on every turn. With Wakka he can use physical attacks or Sleep Attack or Sleep Buster for additional effects. With Lulu as damager, she should cast Demi first to knock Tonberry's health down by a quarter, then proceed with any "ga" spells or Flare.
If Kimahri has an Overdrive, his Bad Breath (obtained by using Lancet on a Malboro) will inflict either Sleep or Confusion on Tonberry. A confused Tonberry is left wide open for Black Magic spells, but if physically attacked it will recover.
If Yuna is strong enough she can just (Grand) Summon Bahamut and kill Tonberry with Mega Flare. Because Bahamut breaks the damage limit by default the damage potential will go beyond Tonberry's max HP.