This came to me while planning out a Cleric/wizard multiclass 5e character with a necromantic bent. If I were to cast Vampiric Touch (which affects a target when I make a melee spell attack on said target during its duration), and then use inflict wounds with the same hand (which requires a melee spell attack to complete) would both damage effects occur?
Continuing to attack through Vampiric Touch requires an action. So does casting Inflict wounds.
From Vampiric Touch (PHB 285)
Until the spell ends, you can make the attack again on each of your turns as an action.
So if you cast Inflict Wounds, you cannot attack again with Vampiric Touch and if you attack again with Vampiric Touch, your action is used and you can't cast Inflict Wounds.
Wax Eagle has it right. Vampiric Touch doesn't inflict the damage whenever you make a melee spell attack. Vampiric Touch allows you to spend an action to make the associated attack. Inflict Wounds takes an action to cast, thereby using up the action you needed to make the Vampiric Touch attack.
I'd imagine that it may (eventually) be possible to come up with an edge case where you manage to get two full-blown actions in a single round, which would allow you to do both, but it's not as simple as casting Vampiric Touch, and then casting Inflict Wounds.
If Wax Eagle's idea of the Fighter's Action Surge ability works out, you would have to be at least 8th level (Cleric 1/Fighter 2/Wizard 5) to make this work, and you would do up to 5d10 + 3d6 damage (if both hit), and regain 1/2 of that 3d6 damage in HP (assuming you hit with the Vampiric Touch attack). That would also cost you two of your three 3rd level spell slots.