Briseis, the sister/lover, is closely associated with mommy, Thetis; she represents a world of the feminine, the world of childhood, a time when the young man could run to mommy or the shelter of the beloved for comfort. She is stripped from Achilles by the brutal leader, Agamemnon and her seizure precipitates his movement into the next stage of life. Achilles' grief is for more than just a slave girl; it is for the whole world of comfort and easy love that childhood embodies. He grieves for the sorrow of losing himself, losing his innocence.
Patroclus, the brother/friend, is closely associated with the thumotic youth of Achilles. He only appears midway through the epic and his appearance shows a new love for Achilles. Much has been made of the homosexual element btwn Achilles and Patroclus, but I think that is a red herring. Whether he is or is not the lover of Achilles seems inconsequential. Just as Briseis represented more than just a slave girl/concubine, he represents a person loved. His connection to Briseis is highlighted when she, weeping at his death, claims that he was always gentle to her. Patroclus, it seems, embodies all that is best and noble in Achilles - he is the brother figure and the mirror image of the great warrior without the hubris that clouds Achilles' judgment. He is not entirely representative of innocence for he sees the brutality of war, even sees the brutality of Achilles, but he is courageous enough to take action as a young man might, to tend to the wounded, to speak against Achilles' headstrong pride, to want to right through mere force of arms the wrongs done his comrades. When he dies, that other world of youth and happiness, carefree living, the bright future free of cynicism and the weight of adult knowledge, is lost to Achilles.
Priam/Hector represent the third doppleganger. Being that they are father/son they consist as a set, each representing a different aspect of the same thing. Priam is the aged father, looking down the dark tunnel of ruin and despair. Hector is the noble man, already dead - he is Patroclus in nobility and death and he is Patroclus' killer. Yet both also represent the future of Achilles. The Achaean knows, now that he has chosen war over peace, that his demise is not far off. He knows his time is marked and the knowledge, previously remote, is grievous to him. He hearkens back to his father and sees in Priam the sorrow that his own father will experience when he dies. Thus another connection is made between Achilles and Hector, Peleus and Priam. In the last book, Priam becomes the father, figuratively, of Achilles and Achilles trades roles with the now dead Hector. Thus, the realization of age, ruin and death are all made quite immediate to the Greek warrior and he must face the harsh reality of adulthood. He will lose what he loves most, not comfort & joy, not youth & nobility, but his very self. Thus Priam is also representative of "the enemy" - the other which we seek to shut out from our own imagination. We do not wish to think about our own ruin and death, but Troy will fall and Priam will die, slipping in the blood of his own son. Though Achilles will, by then, not be alive, such ruin has been the telos of all his actions to this point. Neoptolemus, his son, will be the killer of Priam (as we learn from Virgil) and thus Achilles will have "killed" Priam - the king within himself. This is a harsh and yet necessary reality and Achilles acceptance of it, represented in his going down into the dust to communally weep with Priam, marks the warrior's capacity for humanness.