character", unlike characteristic, is never used to describe a single feature. You can say things like "This man/woman/place has character, and the word here refers to all the features that makes this man, woman or place interesting and different. When you're refering to a single trait or feature that someone or something has use "characteristic".
Besides, "character" is normally uncountable whereas "characteristic" is countable so the "a" in your example gives a pretty good clue as to which of the two words should be used,