George R. R. Martin has commented himself that Bran the Builder may have just been a mythical figure.
If time is permiting would you mind giving a brief description on how the wall was constructed?
Much of those details are lost in the mists of time and legend. No one can even say for certain if Brandon the Builder ever lived. He is as remote from the time of the novels as Noah and Gilgamesh are from our own time.
So Spake Martin, The Wall
However, if he is to have existed this is what we know of him from an in universe point of view.
He built the Wall and Winterfell
Thousands and thousands of years ago, Brandon the Builder had raised Winterfell, and some said the Wall.
A Game of Thrones, Bran IV
He likely built other castles
"No," Ned said. He saw no use in lying to her. "Yet someday he may be the lord of a great holdfast and sit on the king's council. He might raise castles like Brandon the Builder, or sail a ship across the Sunset Sea, or enter your mother's Faith and become the High Septon." But he will never run beside his wolf again, he thought with a sadness too deep for words, or lie with a woman, or hold his own son in his arms.
A Game of Thrones, Eddard V
He lived in the Age of Heroes
Jon had never heard this tale before. "Which Brandon was this supposed to be? Brandon the Builder lived in the Age of Heroes, thousands of years before Bael. There was Brandon the Burner and his father Brandon the Shipwright, but—"
A Clash of Kings, Jon VI
He gave the Night's Watch the Gift
"The Night's Watch," he answered. "This is the Gift. The New Gift, and north of that Brandon's Gift." Maester Luwin had taught him the history. "Brandon the Builder gave all the land south of the Wall to the black brothers, to a distance of twenty-five leagues. For their . . . for their sustenance and support." He was proud that he still remembered that part. "Some maesters say it was some other Brandon, not the Builder, but it's still Brandon's Gift. Thousands of years later, Good Queen Alysanne visited the Wall on her dragon Silverwing, and she thought the Night's Watch was so brave that she had the Old King double the size of their lands, to fifty leagues. So that was the New Gift." He waved a hand. "Here. All this."
A Storm of Swords, Bran III
He sought help of the Chldren of the Forest and learnt their speech
Their song and music was said to be as beautiful as they were, but what they sang of is not remembered save in small fragments handed down from ancient days. Maester Childer's Winter's Kings, or the Legends and Lineages of the Starks of Winterfell contains a part of a ballad alleged to tell of the time Brandon the Builder sought the aid of the children while raising the Wall. He was taken to a secret place to meet with them, but could not at first understand their speech, which was described as sounding like the song of stones in a brook, or the wind through leaves, or the rain upon the water. The manner in which Brandon learned to comprehend the speech of the children is a tale in itself, and not worth repeating here. But it seems clear that their speech originated, or drew inspiration from, the sounds they heard every day.
The World of Ice and Fire, Ancient History: The Dawn Age
He may be a legend
Names such as Brandon the Builder, Garth Greenhand, Lann the Clever, and Durran Godsgrief are names to conjure with, but it is likely that their legends hold less truth than fancy. Elsewhere, I shall endeavor to sift what grain can be found from the chaff, but for now it is enough to acknowledge the tales.
The World of Ice and Fire, Ancient History: The Age of Heroes
He raised Storm's End, is the founder of the Starks and is probably a conflation of many Starks
It's worth highlighting the part in the below quote that tells us Brandon the Builder could be a mix of multiple historic Starks/Brandons merged into one to create an even more legendary figure.
The greatest castle of the North is Winterfell, the seat of the Starks since the Dawn Age. Legend says that Brandon the Builder raised Winterfell after the generation-long winter known as the Long Night to become the stronghold of his descendants, the Kings of Winter. As Brandon the Builder is connected with an improbable number of great works (Storm's End and the Wall, to name but two prominent examples) over a span of numerous lifetimes, the tales have likely turned some ancient king, or a number of different kings of House Stark (for there have been many Brandons in the long reign of that family) into something more legendary.
The World of Ice and Fire, The North: Winterfell
He was a descendant of Garth Greenhand
Brandon the Builder was descended from Garth by way of Brandon of the Bloody Blade, these tales would have us believe, whilst Lann the Clever was a bastard born to Florys the Fox in some tales or Rowan Gold-Tree in others.
The World of Ice and Fire, The Reach: Garth Greenhand
He designed the fifth tower
It was only with the building of the fifth tower, the first to be made entirely of stone, that the Hightower became a seat worthy of a great house. That tower, we are told, rose two hundred feet above the harbor. Some say it was designed by Brandon the Builder, whilst others name his son, another Brandon; the king who demanded it, and paid for it, is remembered as Uthor of the High Tower.
The World of Ice and Fire, The Reach: Oldtown