To expand on the answer provided by AncientSwordRage, the backstory was retconned into Diablo 2; since there was very little provided in the first game. However, there were a few small hints (that Blizzard used to tie the story created in Diablo 2) that can be found in Diablo 1.
Firstly, King Leoric, and his son Albrecht were the first victims of Diablo. When you receive the "Curse of King Leoric" quest from Ogden, he reveals the link between Leoric's family and Diablo.
The village needs your help, good master! Some months ago King Leoric's son, Prince Albrecht, was kidnapped. The King went into a rage and scoured the village for his missing child. With each passing day, Leoric seemed to slip deeper into madness. He sought to blame innocent townsfolk for the boy's disappearance and had them brutally executed. Less than half of us survived his insanity...The King's Knights and Priests tried to placate him, but he turned against them and sadly, they were forced to kill him. With his dying breath the King called down a terrible curse upon his former followers. He vowed that they would serve him in darkness forever...This is where things take an even darker twist than I thought possible! Our former King has risen from his eternal sleep and now commands a legion of undead minions within the Labyrinth. His body was buried in a tomb three levels beneath the Cathedral. Please, good master, put his soul at ease by destroying his now cursed form...
Additionally, Ogden's greeting implies that the hero has "returned".
Thank goodness you've returned! Much has changed since you lived here, my friend.
I believe this was only for the Warrior, implying that he was the only one of the three that came from Tristram, but I can't find any citation for it at this time
The only other link we might be able to identify is that, while you never see the character's face (it is silhouetted in the animation) we have 2 clues:
- The voice (when they scream after shoving the Soulstone into their head) is male
- The armour looks like the Warrior's late game armour.
So, this gives us a link between the Warrior, and the Dark Wanderer/Aidan within the context of the first game.
That leaves the other two; the Rogue, and the Sorcerer. Unfortunately, there is no link to these two within the context of the first game. But, as the previous answer provides, they were addressed in Diablo 2. The lore also expands on this, adding more detail about the characters themselves.
Moreina was the Rogue from Diablo 1.
Blood Raven is the canonical depiction of the Rogue class in Diablo I, though remains unnamed in said game.
The lore expands on her situation, following the events of Diablo 1. The evil within the Cathedral was more powerful than anyone expected, being powerful enough to corrupt any who ventured long, and deep enough into the dungeon.
After Diablo's defeat, Moreina returned to her order. However, she carried something dark within her; a silent madness that ate away at her once noble heart. […] taking on the name of "Blood Raven," Moreina fell into league with Andariel as the Lesser Evil took over Eastgate Keep. Blood Raven assaulted her fellow Rogues in the battle,1 and at some point, was made undead.
Jazreth is managed similarly in the linkage between the games.
The Summoner is the sorcerer character of the first game, though remains unnamed in said game. The confirmation of the two being the same character came much later.
Similar to Moreina, Jazreth was also affected by his delving into the Cathedral depths, but instead of being corrupted by the evils, it instead drove him mad; he hunted for knowledge, and power.
Scarred by his experiences, Jazreth traveled to Lut Gholein to find the mystical Arcane Sanctuary of the great mage Horazon. He met Drognan, an old comrade of his, and told him of his fight against Diablo, but such were his ramblings that the man dismissed much of what he had said. […] Drognan would later speculate that the Summoner had possibly been possessed by the spirit of Horazon.
The Summoner believed that he could bend evil forces to his will. What he didn’t know was that evil uses man, not the reverse.