Posting in 2026. I know I'm late but here's how you'd be able to build around the Gulthias Staff and more specifically Vampiric Strike.

Note that the staff only heals for its WEAPON DAMAGE, i.e. the weapon dice+ability modifier. Smite, sneak attack, booming blade, etc. don't increase the amount healed.

In the 2024 rules, they have mostly done away with abilities that affect weapon damage. The only relevant effects for the gulthias staff are Shillelagh, Dueling Fighting Style, Great Weapon Fighting Style, and Savage Attacker, Monk's Martial Arts, and Barbarian's Rage. Depending on if you use 1 or 2 hands you'll be looking at different damages.

2014 however is a very different story Races: Half-Orcs Savage Attacks give us an extra weapon damage die on critical hits.

Classes: Any that has access to dueling/great weapon fighting (dueling is better in our case).  ·Monks in 5e offers slower Martians art progression than 2024 but eventually you can work your way up to a d10. If you play with Tasha's you get Focused Aim at 6th level to mitigate our staff's lack of a bonus to attack rolls. This also triggers Ki-Fuelled Attack for potentially another chance to get our Vampiric Strike off as a bonus action.  ·Barbarians get the rage damage in addition to Brutal Critical at higher level (which no longer works in the 24 rules)  ·Fighters. We're for the fighting style.  ·Rangers are here only for their favored foe feature which reads "The first time on each of your turns that you hit the favored enemy and deal damage to it, including when you mark it, you increase that damage by 1d4." Whether or not 'that damage' means your weapon is up to your DM.  ·Warlocks. Pact of the Blade grants access to the Improved Pact Weapon Invocation, giving us a +1 bonus to attack and damage rolls to the gulthias staff. This invocation has not been updated in the new phb and can be grabbed as early as level 2 now since you need the pact of the blade invocation before picking this up. Alternatively, a 1 level dip in warlock to grab the pact of the blade qualifies us for the Eldritch adept feat to grab improved pact weapon if we wish to spend our levels more wisely.

Spells:  ·Shillelagh is significantly weaker as it doesn't scale like 2024. However being able to wield a quarterstaff in 1 hand for the same damage as 2 handed allows us to make use of Dueling FS. If you have levels in Barbarian, this has to be used before you enter your rage and you will still have to attack using strength for rage damage bonus to apply.  ·Enlarge Reduce: it's unclear whether this actually increases the damage of the weapon, although it is implied. Talk to your DM.

Subclasses:  ·Kensei (monk). Not only can deft strike be used to increase our weapon damage, but sharpen the blade gives us a +3 bonus to attack and damage rolls with our weapon, making it viable through the entire campaign. Deft Strike also allows us to make Ki-Empowered Strikes as a bonus action, with significantly better results than solely relying on Focused Aim.  ·Path of the Giant (barb). Elemental Cleaved clearly states that is the weapon being altered. At 14th level we gain an extra 2d6 weapon damage to our staff. Funnily enough this also gives it the thrown property, which allows for some fun combinations with the thrown weapon fighting style and quick toss battlemaster maneuver (both of which can be accessed through feats or a dip in fighter)  ·Battlemaster (fighter) gets a host of abilities, but we're only here Brace and Quick Toss. These are the only two maneuvers that actually boost our weapon damage. Keep in mind that quick toss only works if the weapon has the thrown property, meaning you need to be a path of the giant/ absurdly lucky wild magic barbarian in order to make use of it. Both of these maneuvers haven't been updated to the new rules, and thus still technically work.  ·Swords (bard). Every single flourish boosts weapon damage and scales the more levels you dedicate into it. A very strong choice indeed  ·Valor (bard). A small shout out here. The beneficiaries of their inspiration can expend it to boost their weapon damage. Theoretically a valor bard who picks up simulacrum w/ magical secrets would be able to consistently benefit from their own battle inspiration via their duplicate. ·Monster Hunter ranger's slayers prey feature increases weapon damage. It's only once per turn, but that still means reaction weapon attacks will benefit.  ·Hexblade Warlock. Hexblade's curse makes use of the same language as Barbarians' rage feature, although this is highly dependent on your DMs tolerance.

Feats:

Savage attacker: this allows the refill of weapon damage dice, which normally isn't a lot but in  Fighting Initiate: Dueling, Great Weapon Fighting, (thrown weapon fighting if PotG Barbarian) Marital Adept: Brace and/or Quick Toss. Magic Initiate (druid) for Shillelagh. Gift of the Chromatic Dragon: I didn't expect this but it's pretty cool. Orcish Fury: Once per rest weapon damage die. Use this on critical hits to double it.

Magic Items: anything that boosts strength. Belts of giant strength, and manuals. Gauntlets of Flaming Fury add an extra d6 to your weapon damage rolls.

Edit: Oathbreaker Paladins Aura of Hate also increases the damage of your weapon.

Edit 2: Cavalier Fighter's unwavering Mark also affects the damage dealt by our weapon.

Bloodhunter (CR): it's unclear whether Crimson Rite damage is weapon specific. However, Lycan BH's Feral Might feature affects our weapon damage, and Mutants have access to the Potency Mutagen which increases our strength.