Expression for when a person tries to tarnish the relationships of others out of spite or for personal gain

Solution 1:

"out for blood" this an old expression but still commonly used to mean someone is looking for revenge or to stir up trouble for their own satisfaction.

Solution 2:

• slander a malicious, false, and defamatory statement or report:
  a slander against his good name.

You could say XYZ's slanderous attacks on his most loyal ally (friend) proved to be his undoing.

One of the most common ways to blacken the good name of someone is by spreading or circulating rumours and in a…

• smear campaign a campaign to tarnish the reputation of a public figure, especially by vilification or innuendo.
  He repeated his claim that the allegations were the result of a smear campaign.

If we apply it to the OP's example we get:

Rather than leaving gracefully, she contacts former employees and tries to undermine their loyalty to the company and to each other. It could be said that she is running a smear campaign.

Wikipedia says

A smear campaign, smear tactic or simply smear is an effort to damage or call into question someone's reputation, by propounding negative propaganda. It can be applied to individuals or groups.
Common targets are public officials, politicians, political candidates, activists, and ex-spouses. The term also applies in other contexts such as the workplace.

In politics, it has its own unique expression, the mudslinging needn't be reserved for opponents, it can also be directed against someone from their own party who is running for the same office.

• Negative campaigning