the Player's manual says about spell attacks:
Some spells require the caster to make an attack roll to determine if the spell effect hits the desired target. Your attack bonus with a spell attack is equal to your spell casting modifier + your ability bonus.
So, in the case of Flame Blade, you would use your spell casting modifier + your ability bonus.
There are spells that use a melee attack that are not "melee spell attacks". For example, Booming Blade and Green-Flame Blade (SCAG, pp. 142-143) do not depend on a melee. spell out attack. Instead you must make an attack with a melee weapon. They say:
As part of the action used to cast this spell, you must Make a melee attack with a weapon. against a creature within the range of the spell, otherwise the spell fails.
This would be different from the spell in question because, you are making a attack with a melee weapon, so you would add your modifier that applies to that weapon (Strength or Dexterity) and your ability (if you are competent with that weapon).