I think one of the most endearing and memorable friendship in the Harry Potter books was the one that existed between Lupin and Sirius. These two individuals were polar opposite to each other and yet their friendship was so strong and unique. In many ways Lupin and Sirius were like yin and yang. While Lupin was calm, Sirius was loud. While Lupin was vigilant, Sirius was reckless. While Lupin was an introvert, Sirius was a complete extrovert. While Lupin was diplomatic, Sirius was tactless. But there was one area in which they were similar, that is they brooded and obsessed over what they could not control. While for Lupin it was the fact that he was a werewolf, for Sirius it was the fact that he wanted to get away from his family legacy and deny it in every possible way. Sirius wanted to escape from his family roots and traditions in every possible manner and this is evident in his actions throughout the entire series.
There is also the parallel between the upbringing of Harry and Sirius. Be it the emotional abuse or a greater attention lavished on a different child (Regulus in case of Sirius and Dudley in case of Harry), the growing years of both Harry and Sirius were painful and traumatic. So while Sirius responded by being rebellious, Harry chose to withdraw into the shadows while at home. While Harry made sure his uncle and aunt ignored him, Sirius made it very difficult for people around to ignore him. This resulted in Sirius not caring for the consequence of his actions until they had already been played out in front of him. So many incidents be it telling Snape how to get past the Whomping Willow, tracking Peter down after that Halloween, mistreating Kreacher, rushing to the Department of Mysteries, taunting Bellatrix, all are proof that Sirius was very hot-headed and emotional. This is a major reason why he suspected Lupin of treachery because he felt that Lupin too would have wanted to defy what he was and become something more valuable and strong. (However, remember that Remus too suspected Sirius, and for the same reason, another moment to show that they are two sides of the same coin.)
After his stint at Azkaban, Sirius becomes mentally ill though he is still loyal and brave. Dementors are J.K. Rowling’s analogue of depression. She describes this emotional state as the “absence of being able to envisage that you will ever be cheerful again. The absence of hope. That very deadened feeling, which is so very different from feeling sad.” In addition, twelve years of being denied any form of human contact is going to affect the sanity of an individual. After his escape, Sirius is so fixed on revenge that he tries to attack anyone who stands in the way. He understands that Crouch mistreating Winky is not right and tells Harry that the manner in which a man treats his inferiors is a reflection of his character but cannot equate the same in his treatment of Kreacher. It’s strongly indicated that he drinks in order to cope and both Molly and Hermione have expressed concern at his difficulty in differentiating between Harry and James. So when in Harry Potter and the Order of Phoenix, he is forced to stay at his childhood home, away from the public eye, it was just a disaster waiting to happen. In a way, I blame Dumbledore for having Sirius stay in that house because it must have been so difficult and traumatic for him to relive his childhood days again. But that being said, Sirius was extremely brave and his loyalty and strength of mind speaks volumes about his character.