I'd like to optimize my character concept:

I want my main damage source to be a Shillelagh'd weapon and squeeze that source for all it's worth.

Table character limitations:

• Character is level 5


• Point buy for level 1 ability scores


• No multiclassing allowed


• Variant Humans and Duergar are banned


• Each character gets a free feat at level 1, provided they qualify for that feat at level 1


• All official sources are allowed and can be mixed, but UA is only allowed on a per case basis (so preferably not used), no homebrew

Optimization Goals:

• Highest average damage per round possible, using Shillelagh


• Secondary goal is highest effective number of hit points, assuming a well mixed pool of damage sources

Bonus points for:

• Not using UA


• Actually having a use for my Wisdom score besides whacking with Shillelagh

So far I have:

• Hill dwarf for the juicy Con and Wis + free HP


• Go Fighter, archetype Champion for extra attack DPR and heavy armor + shield + armored fighting style


• Take Magic Initiate feat level 1 for Shillelagh + absorb elements


• Pump Wis with ASI

Hunter Ranger

Combat-wise, you get:

• Dueling fighting style


• Extra Attack, Hunter's Prey (your pick)


• Spells (like Hunter's Mark)

You also get Favored Enemy, Natural Explorer and non-combat spells, for utility. Assuming you have cast Shillelagh, Hunter's Mark (at level 5, you can do this 6 times a day) and chosen Colossus Slayer, this means you make 3d8+2d6+4+WIS*2 when you land both your attacks on a single-target.

Compared with the Fighter, you lose some AC (medium-armor only), but you may not have disadvantage on Stealth checks, depending on which armor you choose to use. You also lose Second Wind and Action Surge (each once per short rest), and Improved Critical. I haven't done the math for Champion, but I think at level 5 you can out-DPR it as a Hunter Ranger.

As an alternative to a generic anything-with-Extra-Attack-answer:

Draconic Sorcerer 5.

Benefits:
Twinned Green Flame Blade, allowing you to set your stick on fire and bash two people at once for 1d8 (bludgeoning) + 1d8 (fire) + Wis-mod, and if those enemies are adjacent to each other they each take an extra 1d8 + Cha-mod fire.

Draconic Sorcerer gets an increased AC and extra health (stack it with Hill Dwarf to have an unreasonably healthy sorcerer).
Haste will allow you to get another stickbash out every turn, while ensuring your enemies can't flee from your stick.
Shield will give you a ton of survivability to stick around in melee.

The downside of this is a bit of multi-ability-dependency in the build, but there are plenty of sorcerer spells that don't use charisma at all - selecting those (haste, blink, mirror image, blur, etc..) allows you to go all in on the Wisdom-melee style.