The grammatical question in point is using commas with restrictive and nonrestrictive clauses.
Restrictive clauses are those which are critically necessary to identify the noun they're describing. Removing them either changes the sentence and/or it becomes unclear what noun out of many similar nouns the sentence is referring to.
e.g. The woman who's speaking now is my mother. (This is restrictive, because it limits the wide range of potential women who could be my mother to just the one speaking now. Remove it, and you have no idea which woman is my mother. )
Nonrestrictive clauses are those that don't specifically identify the noun they modify, but just give extra bonus information. Remove it, and the sentence is essentially unchanged and still clear.
John Doe, who's the current company CEO, hates cats. (This is nonrestrictive, because although it does give us some extra information about Mr Doe, removing it doesn't change the basic idea of the sentence that he hates cats. Nor is it unclear who hates cats if you remove it.
The rule is that nonrestrictive clauses are separated with commas. Restrictive clauses are not.
Regarding the original sentence:
Martha is a bright girl who displays good character.
Personally, I wouldn't consider it restrictive. Martha's still a bright girl, regardless of whether she has good character, bad character, or no character. Her character is just bonus information about her, but the original sense of the sentence about her intelligence is still unchanged.
Ergo, I'd use the comma.
Martha is a bright girl, who displays good character.