When you make a melee attack with your pact bow, it counts as an Improvised Weapon that you are proficient with, uses Str or Cha (your choice) as the modifier, and all the pact features still apply, but your DM may rule otherwise.

From the Player's Handbook section on Weapons (chapter 5):

If you use a weapon that has the ammunition property to make a melee attack, you treat the weapon as an improvised weapon.

The bow has the Ammunition property and you are making a melee Attack with it.

According to the rule on Improvised Weapons in the same chapter, the bow deals 1d4 damage if you hit:

If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage.

You are proficient with your pact weapon when using it this way:

See this passage from the PHB entry on the Warlock (Pact of the Blade):

You are proficient with it while you wield it.

You're wielding your bow even though you're making a melee attack, so you're proficient with it. Note that this specific rule may contradict the general rule that characters are not usually proficient with Improvised Weapons, but may be proficient at the DM's discretion:

Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

In D&D 5e, the specific rule "you are proficient with it", trumps the general rule "at the DM's option". The introduction to the Player's Handbook has a section titled Specific Beats General:

Remember this: If a specific rule contradicts a general rule, the specific rule wins.

You use Str or Cha as the ability modifier:

The ability modifier used for a melee weapon attack is Strength, and the ability modifier used for a ranged weapon attack is Dexterity.

Making an improvised melee attack with the bow counts as a "melee weapon attack". So your attack and damage modifier is Strength. However, since this character is a Hexblade, the Hex Warrior feature applies:

Whenever you finish a long rest, you can touch one weapon that you are proficient with and that lacks the two-handed property. When you attack with that weapon, you can use your Charisma modifier, instead of Strength or Dexterity, for the attack and damage rolls. This benefit lasts until you finish a long rest. If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon’s type.

So you can use Cha as the modifier, even with the improvised melee attack.

Lifedrinker, Thirsting Blade, and Improved Pact Weapon all apply:

There is no reason that I can see that your eldritch invocations wouldn't apply. E.g. Lifedrinker applies...

When you hit a creature with your pact weapon

That's unambiguous. If your attack hits, you have in fact hit a creature with your pact weapon.

As for Thirsting Blade:

You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.

That also applies. Ditto for Improved Pact Weapon from XGtE. As the adage goes, "the rules do what they say".

But your DM may rule otherwise:

The DM could rule that if you're using your pact weapon in an improvised way, then it's "no longer your pact weapon" somehow.

See also this tweet from Jeremy Crawford (the lead rules designer for Dungeons & Dragons 5th edition):

If you use a weapon in a way that turns it into an improvised weapon—such as smacking someone with a bow—that weapon has none of its regular properties, unless the DM rules otherwise.

If the DM rules this way, then your pact bow is just an improvised weapon that you're not proficient with, uses Strength as the modifier, and none of the pact features apply.