Is it possible to use Booming Blade as part of my charger feat attack? Or is it separate? If so, what would my attack damage calculation be (assuming I'm using a greatsword and have Great Weapon Master and utilizing the -5/+10?)
The Charger feat description reads, in part:
When you use your action to Dash, you can use a bonus action to make one melee weapon attack or to shove a creature.
—Player's Handbook ch. 6
Booming blade is a cantrip (a spell). Its spell description states, in part:
Casting time: 1 action
As part of the action used to cast this spell, you must make a melee attack with a weapon.
—Sword Coast Adventurer's Guide ch. 4
The Charger feat description does not state that you can cast a spell with your bonus action using this feat. Even though making a melee weapon attack is part of casting booming blade, it is still considered casting a spell. Furthermore, the cantrip's casting time is one action, not one bonus action.
By the rules, a PC cannot cast booming blade using the bonus action granted by the Charger feat.
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Addendum: I reviewed the OP's original question before it was edited by another user. The OP asked if it was possible "to make two attacks, one for booming blade and the other as a bonus attack for charging." The answer is no, not in the same turn, as the Charger feat requires to use your action to Dash in order to be able to attack using a bonus action. Of course a fighter using Action Surge would be the exception to this.