Lip's intelligence is the biggest source of his unhappiness: he is a brilliant student (he has been offered a place on two universities with scholarship), yet he is afraid of leaving behind his past, his family and friends. This fear is best seen when he is offered a place at the West Point, something that his brother Ian dreamed about. Ian says:
I'm sick of living in your shadow.
Lip then became convinced that soon everyone will be living in his shadow, that everyone that he cares about will hate and envy him. This leads to his continued self-handicapping and self-loathing - just look at his tendency for self-harming: whenever he is upset, he picks a fight with someone that is bigger and stronger than him.
Edit after watching to the latest episodes
I think THE main issue with Lip is shown during his hearing in front of the University Board when he says:
My father has a brilliant mind and I believe so do I, but I don't want to be like my father.
Phillip can see, that he is slowly becoming Frank: yes, he is smart, but he is also falling into alcoholism and has anger-management issues. Who wouldn't be constantly unhappy living with someone like Frank and know, that it is quite probable that you will be just like him one day?
Another topic is his problem with women: he is so afraid of having a normal, monogamous relationship in which he could engage emotionally that he sticks to having semi-causal "f@cking-buddies", but at the same time you can see that he indeed longs to love someone... so funny enough he chooses loving kids, because loving a child is (usually) easier than loving an adult.
Lip has many demons that he has to face every day.