He doesn't care about speaking the truth. (in the sense that he's gonna tell the truth, no matter what).
If I translate it directly, it is going to be かれは 真実を 話すことに ついて 気にしない。
In more natural Japanese, かれには 真実を 話すかどうかなど どうでもよい。
He doesn't care about his friends.
かれは ともだちの ことを 気にかけない。means ともだちの ことなど どうでもよい。
He doesn't care about studying.
かれは べんきょうすることに かんしんが ない。
I think "He / She doesn't care about ~" can be translated in some Japanese meanings, depends on the situations.
Hope it helps you.