The internal conflicts and external conflicts in The Picture of Dorian Gray are almost interconnected.
The main conflict is moral negotiation, and this affects both the internal and external incidents in the novel.
The first incident is the first meeting between Dorian and Henry. The revealing topics spoken during that first meeting basically opened Dorian's eyes to the world of hedonism: the eternal search for pleasure and for the enticement of sensations. The first conflict begins: can Dorian reconcile his youth and beauty with the fact that such things are ephemeral? Can he accept the fact that the physical world to which Lord Henry wants Dorian to become addicted is actually a world that is subject to less fashionable variables, such as age, health, time, and death? It is clear that Dorian's psychological and moral make-up renders him a man too easily weakened by the fear of losing his beauty and youth. Hence, his first conflict is acceptance of reality versus going as far as to commit moral suicide: to exchange his soul for eternal youth. This is both an internal and external conflict: the physical versus the spiritual; body versus soul
The second conflict is external: the acceptance of consequences. Dorian basically gives up his soul, and becomes an ageless male. However, what happens to the rest of the world? The rest of mankind will continue its regular lives, aging, getting ill, and dying. Old bets will continue to be on the table, such as the mysterious case of Alan Campbell's past with Dorian and James Vane's wishes to revenge his sister's suicide. Dorian has to drag on his sins as far as he lasts in this earth looking and thinking the same way as he did when he committed his worst crimes years before. He has to live with it, and still fear for his life.
The third conflict is social ranking. Basil, Sybil and James are characters that seem to be disposed off easily in the novel due to their low social rank. Basil himself says at the beginning of the novel that the job of the artist is to entertain the upper classes in order to be let in their presence.
You know we poor artists have to show ourselves in society from time to time, just to remind the public that we are not savages.
Hence, perhaps if the novel had no social differentiation among the characters some conflicts could have been avoided such as Sybil Vane becoming so infatuated by Dorian, and Basil being looked down upon in favor of Henry.
Yet, the main conflict is mainly moral- can Dorian look at the picture, which has now absorbed the moral decay and disgusting nature of his soul, and accept what he has done? Can he reconcile his life to his sin, and his sin to his soul? It is clear that the lack of ability to resolve this particular conflict is what ultimate leads Dorian to his demise.