You would probably say "like likes," with the initial like functioning as sort of an adverbial modifier rather than as a true verb. But odds are you wouldn't want to use the phrase at all, unless you want to sound like you're 12 years old.
Like like (which would typically be formatted as "like like") is an informal phrase that typically indicates that the subject has a romantic "crush" on the object. The phrase turns on the ambiguity of the word like, which ordinarily indicates affection or affinity for something ("I like oranges"), but is also commonly used by school-aged children to indicate romantic feelings, often on the part of someone who is afraid to express them to the object of his or her affections ("Jeremy told me that John likes Erin!").
So perhaps I confront John on the playground one day and I ask him how he feels about Erin. Guardedly, John says "I like her." I respond with "So do you like her, or do you like like her?", meaning "Are you just fond of her as a friend, or do you have romantic feelings for her?"
As this suggests, "like like" has a strong connotation of juvenile silliness about it, so you wouldn't want to use it among adults unless you're intentionally trying to be ironic or humorous.