First and foremost, looking at my Laxus it's important to take his family into consideration. I have made one small change to it, which affects both Ivan and Laxus massively.
Porlyusica is Ivan's mother.
While I won't dive into Porlyusica or Makarov, nor how their family ended up the way it did, I will say that this greatly changes Ivan. His childhood and later obsessions is why he is exiled from fairytail, leaving a young Laxus to be raised solely by his grandfather.
Since she's never mention, I believe Laxus' mother either died in childbirth or was a one night stand of Ivan's that left the baby wth his father. Either way, Laxus has never had a very present female role model. With only his grandfather and the older members of fairytail, this explains his cockiness in the earlier seasons of fairytail.
Laxus was a sickly and weak child, like in canon, which is what lead to his lacrima being implanted. Once it scars over, he gets his guildmark.
From here onwards, he's the Laxus we all know and love! Well, mostly! Laxus is forced into a big brother role as younger kids join the guild and much of his dickishness comes from a place of pride in his guild. It's just rather misguided.
Bickslow is much the same, but he and Laxus are much more like brothers. Bicks is born with his ability to see souls, already having control of one by the time he joins fairytail.
Each of the markings on his body represents a soul he controls, so should he gain another one to control he will gain another marking.
Bicks and Laxus train together constantly, going on jobs together as often as they can. Really, Bicks doesn't change much! He's always been a chill dude.
She joins at 11 years old, and immediately takes to the oldest two! They're cool, afterall, and she needs to be just like them.
Like Bickslow, Ever is born with her stone eyes ability. She struggled a lot until meeting Bicks, who taught her how to control the ability along with getting her sunglasses.
Evergreen officially becomes the little sister of the group, tagging along even when they didn't want her to. Anything Laxus asks of her, she'd do, in order to prove her loyalty. Being the strongest trio at a young age gives Ever her incredible ego.
And our final piece falls into place:
The last of the Thunder Legion to join, and the one to give the team their iconic name. Freed joined due to hearing about Laxus and adored him. Freed immediately became Laxus' hype man, making sure everyone knows that Laxus is the best.
Freed had wanted to learn takeover magic, to be able to become something far stronger in order to fit in with his group. However, script magic soon took his interest and his training forced his body to adapt. Scripts appear on his neck and within his eyes, aiding in his writing and reading.
His eye, however, is permanently changed into that of his "demon" form.