It sounds as if the speaker is talking to an enemy whom he has wished to kill for a long time.
"Auld aquaintance" can be translated to "old aquaintance", or someone you have known for a long time but probably is not a friend.
So, the speaker is asking himself if he should simply forget this person, but answers saying "No," and that the question doesn't even actually need a response because the answer is so obviously "no".