From Appendices we can reconstruct this character. First fact: he is an orc chieftain. Such guys are always more capable fighters, stronger, faster, more agile, the elite of the orc race. Their ,,subjects" are fiercely loyal to them (wanting to avenge their deaths). There is not much details about his life, but at one point he took his people to Moria and which is surprising he deemed himself to be an independent king there! That's an ambition :), anyway he wasn't specifically keen on hunting down the line of Durin as movie states. It just happened that Thrór came there with one companion Nar acting proudly ,,as it's rightful heir returning". Azog shows some of his skill and traits in the fact he is literate, he is proud and knows value of money :) he spares Nar and mockingly throws pouch of small money as ,,payment" for him sending message to family of Thrór. This act of murder, desecrating corpse as a general insult to dwarven race initiated the War of the Dwarves and Orcs, so Azog must have been the leader of all orcish population in Misty Mountains with Moria as his seat of power! This implies he was very powerful among his kind.
In The Hobbit he serves as a justification of orc's pursuits and special hatred towards dwarves. His son Bolg wants to avenge him and additionally ,,gain dominion over whole North" other orc communities want to avenge the death of Great Goblin and of their comrades, also to claim the Erebor's tresure for their own. Bolg rallies the orcish tribes at his capital Mount Gundabad to fulfill these goals, his parentage was the reason he so easily assumed control over orcs, he inherited the role of a leader.