I see mainly 2 flaws in the action sequence itself, plus 1 more in the build (at 5th level).
The first round uses two bonus actions. This would work in my world, since I allow characters to use a bonus action for their main action, but RAW this is not allowed.
Second round: the Booming Blade only applies to the first action. On his Action Surge, he can either take the two swings, or cast another Booming Blade and take one swing, but not both. And if he uses the Booming Blade again, he has not taken an attack action, and thus won't get the offhand attack with the 2nd weapon.
So the total roll becomes: 5D8 + 7D6 + 12, for an average of 59 hp of damage, assuming every swing hits. Plus more if the BBEG misses his save on the Fire Rune, and/or moves and takes more damage from the first Booming Blade.
The build is almost possible. It's a very effective build, for its purpose, and is mostly legal and according to the RAW.
He has to have a 16 or 17 Strength, which he could do if he is a mountain dwarf. He would have to be a 4th level Rune Knight, with the Fire Rune, the Two-Weapon fighting style, and the Dual wielder feat taken at 4th level fighter.
BUT he would have to have multiclassed into Warlock for 1 level, to get Hex and Booming Blade, in which case, he doesn't have Extra Attack yet (but would get it next level). Or, if he has Extra Attack, he has no way to have gotten the spells, unless the DM has given him a free extra feat for some reason. So at a total of 5th level, he is doing too much, but next level he could have pretty much all of it.
So really, he is doing a little bit more than what is possible under RAW, but not a huge amount. It's a very good build, at that one thing. The main thing to remember, though, is that he's a one-trick pony. He can do that one thing well, if he can get to the BBEG with his 25' move and no special mobility abilities, past all minions and obstacles, and hasn't used up his Action Surge, Fire Rune, Giant's Might, or single spell slot before getting there.
And even then, this is a once-per-short-rest event -- even in this one fight, he has to set up for a round before he can do this, and then gets one great round, but has then used all that stuff and is back to pretty much being a regular fighter. Action Surge, Giant's Might, the Fire Rune, and his spell slot are all one-shots until a rest is taken. After that, his third and subsequent rounds look like this (assuming 6th level total, so he has gotten both the Warlock bits and the Extra Attack):
Turn 3
Action: Cast booming blade
Main hand Melee: 1D8 +3
Booming Blade: 1D8
Hex: 1D6
Giant’s Might Passive: 1D6
-- OR --
Action: Attack with melee weapon (Extra Attack gives two attacks)
Main hand Melee: 1D8 +3
Hex: 1D6
Giant’s Might Passive: 1D6
Main Hand Melee: 1D8 +3
Hex: 1D6
Bonus Action: Two weapon fighting attack
Melee weapon attack: 1D8+ 3
Hex: 1D6
Total roll: 2D8 + 2D6 + 3 = Avg 19 hp of damage with the Booming Blade, or
3d8 + 4d6 + 9 = 35 hp of damage (with regular attacks, if they all hit), which is pretty good, but not devastating.
To compare, a 5th level wizard has two fireballs, each for 8d6 (Avg 28 hp of damage) to multiple targets from a distance -- he doesn't even have to get anywhere near the BBEG, and clears out a bunch of the minions at the same time.
Or a 5th level Vengeance Paladin, who looks like this (using the same race, Great Weapon fighting style, and GWM feat:
Turn 1
Bonus Action: Vow of Enmity (gives advantage)
Action: Attack with melee weapon (Extra Attack gives two attacks)
Two hand Melee (w/ GWM): 2d6 + 15
Smite (2nd lvl slot): 3d8
Two Hand Melee (w/ GWM): 2d6 + 15
Smite (2nd lvl slot): 3d8
Turn 2:
Bonus Action: Hunters Mark
Two hand Melee (w/ GWM): 2d6 + 15
Hunters Mark: 1d6
Smite (ist lvl slot): 2d8
Two Hand Melee (w/ GWM): 2d6 + 15
Hunters Mark: 1d6
Smite (1st lvl slot): 2d8
total roll 10D8 + 10D6 + 60 = Avg 140 hp damage (plus rerolling 1s and 2s, which is good for another 10 to 20 hp of damage)