On the first turn, Quicken Spell will let you attack twice with it. By casting it as a bonus action (which includes attacking with it), you can free up your action, allowing you to attack with it again.
Other than that, the Fighter's Action Surge ability is your only option here.
Until
the
spell
ends,
you
can
make
the
attack
again
on
each
of
your
turns
as
an
action.
While there are various methods of speeding up the casting of a spell, Vampiric Touch gives you an extra use for your action. As such, only abilities that let you make multiple actions, not multiple attacks, can help you use it more than once. The fact that it involves a melee spell attack is unimportant.
As you've said, Haste won't work, since it specifies what actions can be used with its extra action. Time Stop won't work either (well, it will let you use Vampiric Touch on yourself, if that was a thing you wanted to do). Action Surge is the only ability that will give you an extra action with no restrictions.
Note that even with Action Surge, using it more than once in a turn is still potentially shaky. It says you can make the attack again on "each of your turns", which could be taken to imply that you can only use it once on each turn.
Yes, but you have to be level 12 to do this
The Sage Advice articles says:
You can also use Pact of the Blade to bond with a magic weapon, turning it into your pact weapon. This magic weapon doesn’t have to be a melee weapon, so you could use the feature on a +1 longbow, for instance
This takes care of the ranged pact weapon, and allows the use of both Thirsting Blade and Lifedrinker with a ranged weapon. This means you can attack twice with your Attack action using a magic ranged weapon.
And the Sage Advice compendium says:
Does Crossbow Expert let you fire a hand crossbow and then fire it again as a bonus action? It does! Take a look at the feat’s third benefit. It says you can attack with a hand crossbow as a bonus action when you use the Attack
action to attack with a one-handed weapon. A hand crossbow is a one-handed weapon, so it can, indeed, be used for both attacks, assuming you have a hand free to load the hand crossbow between the two attacks.
This allows you to attack twice -- once with your action, another with your bonus action. Since you have Thirsting Blade (which works with any weapon as long as it's your pact weapon), you can get three attacks in all.
However, while Thirsting Blade is available at 5th level, Lifedrinker is available at 12th level. So to complete this build, you have to be a 12th level character at minimum. Most of it, you can enjoy from 5th level onwards though, as only Lifedrinker is the high level component.
Best Answer
No, you can't improve Thirsting Blade with Vampiric Touch
Thirsting Blade is a Warlock's counterpart of the Extra Attack feature:
Vampiric Touch is a spell involving a spell attack:
When you cast Vampiric Touch, firstly you take the Cast A Spell action and expend a 3rd level spell slot. Then you deal 3d6 and only 3d6 necrotic damage, if you succeed with the melee spell attack. This melee spell attack is a part of the spell — you literally have to touch the target with "your shadow-wreathed hand" to apply the spell effect.
Within 1 minute, you can repeat this action without expending more spell slots, providing you maintain the concentration. This requires an Action. So you have to choose — either you make two weapon attacks with your pact weapon (second one because of the Thirsting Blade invocation), or you make one spell attack with Vampiric Touch.
See also: Can you apply the Vampiric Touch and Shocking Grasp damage together in one attack?
5e is quite restrictive regarding damage output. For instance, you can't stack Thirsting Blade with Extra Attack, if you have the latter from multiclassing:
There are methods of increasing the Vampiric Touch damage output, but they are based on increasing of number of actions, not just number of attacks being made.