Spell attacks use your spellcasting ability
The Player’s Handbook says about spell attacks:
Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target. Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.
So in the case of Flame Blade, you would use your spellcasting ability modifier + your proficiency bonus.
...But you should read the wording carefully
There are spells that use a melee attack that are not “melee spell attacks”. For instance, Booming Blade and Green-Flame Blade (SCAG, p. 142-143) are not dependent on a melee spell 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 one creature within the spell’s range, otherwise the spell fails.
This would be different than the spell in question because, you are making an attack with a melee weapon, so you would add your modifier that applies to that weapon (Strength or Dexterity) and your proficiency (if you are proficient with that weapon).