Why did Jack kill this character?

Is this because he snapped in season 8 and is no longer averse to revenge killing?

Madam President, I don't want revenge, I want justice.
_{-Jack Bauer, Day 8, 9:48pm}

Really I think it's a combination of factors:

1. Jack loves The Hellers
   
   Jack started working with them sometime between the end of Day 3 and the start of Day 4 when James Heller is the Secretary of Defence and Jack is his adviser/bodyguard and Audrey is both her father's secretary and Jack's lover. Aside from their brief falling out in Day 5 when
   

   Jack gets evidence that President Charles Logan ordered the assassination of David Palmer, Tony Almeida and Michelle Dessler and Heller tries to negociate with Logan instead of just calling the Attourneys General

   
   they've always had a close relationship based on trust.

2. Jack loves America, and the President is America

   Jack doesn't play politics, even when he dislikes the president, he'd never put himself above America. Some specific examples:

   • When he captures Logan at the end of Day 5, he doesn't kill him, because bringing him to justice is more important, even though he could easily get away with it.
   • During Day 8, when President Taylor turns against him, he never goes after her.
   • Even later in Day 8, he has the opportunity to kill former President Logan twice, first time when he's in the tunnel and draws them out of their car with smoke grenades, then again later on when he has a sniper rifle pointing at him.

3. Jack always kills the bad guy

   Always. Well, no, mostly.

   Day 1, Victor Drazen runs out of ammo in a gunfight with Jack and surrenders:

   

   Jack shoots him 12 times in the chest and watches in silence as his corpse floats away:

   

   Day 2, he tried very hard to kill Peter Kingsly, but got a last minute assist from a helicopter-based marksman as he'd run out of ammo and was severely injured from beating several mooks to death. He did however encourage his daughter to murder an unconscious man.

   Day 3, Jack's having a bad day, and to make himself feel better he kills an unarmed Nina Meyers after she refused to learn that killing Jack's family is not okay:

   

   Day 4, maybe Jack's calming down in his old age. Today he drops Habib Marwan off a roof, but to be fair he was cutting Jack's hand with a hunting knife, so we'll let that one drop.

   

   Day 5, after Jack is done saving America from Nerve Gas, Dawn Brigade has broken into a submarine and are about to attack America with it. So Jack strangles a guy to death with his murderous thighs:

   

   Not satisfied with this, Jack gets a bit nostalgic and sets up the questionable Christopher Henderson by giving him an unloaded gun. Once Chris attempts to shoot Jack, he fills him fulla lead:

   

   Day 6, after Jack subdues and ties up Abu Fayed in chains, jack activates a pulley system so he's strangled to death:

   

   Day 7 is the major exception. Jack kills Juma in genuine self-defence (he goes for his gun after the White House attack and Jack takes him down). Most of the bad guys take each other out except for Tony, who breaks our hearts and lives.

   Day 8 and the guilty are punished. Jack takes out an unarmed Dana Walsh:

   

   tortures and dismantles Pavel Tokarev:

   

   and turns Mikhail Novakovich into a kebab:

   

So to go back to the original question:

This seemed very out-of-character for Jack.

I very strongly disagree.

Then there's also an out-of-universe reason, it could be actor allusion. It's certainly the first thing I thought of when I saw the scene:

In Game of Thrones, Michelle Fairley plays Catelyn Stark, mother of Robb Stark and in season 3, her son is murdered in front of her in the home of the Freys, whose sigil is the twin towers, before she is murdered herself. In 24, her son is also murdered in front of her as he is thrown out of a tower block window, before Jack does the same to her.