In no particular order then are a brief list of the key psychological themes I picked up upon:
1. Depression. The 'dementor' creatures personifying depression and sucking all joy from those they possess. Chocolate helps after a brush with the dementors!
2. Loss, coping with death. Harry lost his parents aged one following a magical double murder. He spends the books coming to terms with this loss but also suffers the loss of various loved ones throughout the series. There are hard times for Harry but one of the best things about the character is his ability to carry on and not be sucked down by it all, compared to other book characters who react in different ways.
3. Bullying. Another core theme. Draco Malfoy and his cronies are the school bullies, while in the adult wizarding world Lord Voldemort rules by being a hideous bully to those around him, actually bullying Draco's own father into supporting him. Professor Snape was a teacher who tormented Harry for years, although he did have another agenda.
4. The importance of friends. No matter what happens to Harry he always has his two best friends, Ron and Hermione, to fall back on who share in his troubles and help him any way they can. Without family as such, where would Harry have been without his support network of friends?
5. The Power of Love. Dumbledore said it was Harry's main advantage over Lord Voldemort, that he could experience and understand it while Voldemort could not. Professor Snape's entire life was altered by his love for Lily Potter, while Lily's love for her infant son protect him from the killing curse. I think the books were as much about love as they were death and loss in the end.
6. Psychopath. Lord Voldemort is a powerful dark wizard who is essentially a psychopath. He does not care for the emotions of others and does not bat an eyelid before casting his killing curse. He craves power and control and will stop at nothing to get it. But as Dumbledore said, his inability to experience what others feel ended up being his weakness, as he was unable to kill Harry via the protection of his mother.