Ornstein Has a Chronological Story Between Dark Souls 1, 2, and 3. Here it is.

The following was originally posted in this thread by Ornstein_n_smough, and all content contained is credited to them. Only minor changes were made to accommodate the difference in format. 

This post explores the story of Dragonslayer Ornstein.

via darksouls.wikidot.com

So we learn that ornstein is one of 4 knights of Gwyn in Dark Souls 1. He helped defeat the dragons and was given the name of dragonslayer for his deeds. In Dark Souls 2 we fight the old dragonslayer. We never know why, and it always felt like stupid cheap Fan service until now. This is Ornstein's full story, as Vaati would say, prepare to cry.

It all starts out with Gwyn and everyone involved in defeating the ancient dragons. After the defeat of the ancient dragons, I can picture everyone celebrating. Except for one person who was the firstborn. He felt bad for what Gwyn and the other lords had done, so he abandoned his family to be with the dragons. This angered Gwyn so much that he erased his firstborn from the annals of time. This must have taken quite a toll on Ornstein as we find out he was being mentored by the firstborn(who after reading a great theory, it's pretty much perfect sense to call the firstborn Faraam) moving on. Ornstein, who at this point is crushed by the leave of his mentor, can't stand around without him. So Ornstein sets out to look for Faraam the God of war. However, this is no simple task, and it definitely doesn't happen over night.

Enter Dark Souls 2. We find ornstein in Dark Souls 2, corrupted by the abyss, yet still searching for his old mentor. Seeing as how ornstein is completely optional in dark souls 2, it could be proper cannon that our character never found Ornstein. Ornstein existing in dark souls 2 is simply explained by game mechanics. He needed to be there so we could one day find out his story. Like I said before, when we find Ornstein he is corrupted by the abyss. I believe it's at an early stage though as he still knows he was a dragon slayer. When we see him, he is still in progress of looking for Faraam.

Enter Dark Souls 3. After beating Faraam, we finally see what became of Ornstein. He is dead and only his armor remains. I believe Ornstein was too corrupted by the abyss to be saved by the time he finally found Faraam. So Gwyn's firstborn only had one option, he had to kill Ornstein.

EDIT: if anyone is interested in reading about how the firstborn and Faraam from dark souls 2 are the same person, here is the link to the post I read talking about it. It's extremely interesting and more than likely 100% accurate.https://www.reddit.com/r/darksouls3/comments/4h4q96/theory_the_nameless_king_may_not_live_up_to_his/

I would also like to include a comment from the thread, which offers another timeline of events by Salamatiqus:

For me it is different story.

1. Ornstein left to search for Firstborn. He is replaced by illusion to help Smough defend Anor Londo.
2. Appearance in Dark Souls 2 could be explained same way as Dragonslayer Armor. It is just armor with no one inside, a golem. Or, just like OP said, he stumbled upon Drangleic in search of Firstborn, likely heading to Forossa.
3. If I recall correctly, there is no corpse, just white sphere we loot to receive dragonslayer armor. That means he stripped it from himself and dropped it. Which is required to ... use Dragon torso stone!

Once Dragonslayer, now Dragon himself. He found Firstborn, he found peace, he found path of the dragon.

I want to believe.