This means that he does not like chronically invalid people and is determined not to be one of them. Character here basically means personality and a disagreeable character is a personality that is undesirable. The reason 'of' is in the sentence is because the sentence is talking about the character of (belonging to) the chronically invalid.

The 'of' is there because this is a possessive form — the character of a chronic invalid. No character so disagreeable as that [character]. It means he found that kind of personality unpleasant.