This is an excellent question, for two reasons:
1. It is a controversial question, without a definite answer. So it is interesting.
2. It says something about life: that you can take it at the most superficial level and still be happy, or you can take it at a richer and deeper level.
The general superficial man-in-the-street level:
- "Who" is used with a person. "That" is used with a thing. So obviously "who" would have potential to be correct, and indeed in oral English people often use "who" for your sentence. The relative pronoun for this sentence, however, replaces the object (the word him) and not the subject of the original "Somone likes or know him", and so it has to be "whom" to be correct.
The richer, deeper and intellectually exciting level:
- Some grammarians insist on this rule: "Who" should refer to particular persons, while "that" refers to generic persons. And also this rule: "That" is the only defining (as opposed to non-defining) relative pronoun.
- So accordingly, "that" is without doubt absolutely the correct word for your sentence.