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

Wisdom is one of the Ranger's main stats, you get martial features that are very strong at this level, as well as very strong utility in the form of spells and abilities. 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 Booming Blade, allowing you to charge your stick with thunder and bash two people at once for 1d8 (bludgeoning) + 1d8 (thunder) + Wis-mod, and if those enemies move later an extra 2d8 thunder.

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.

hexblade bladelock, something that gets +cha

Hexblade lets you attack with charisma. Quarterstaff is versatile rather than two-handed, so it's legit (you won't be using the versatile bit. It's redundant).
For invocations, take Thirsting blade at level 5, giving you two attacks per attack action.
Take Polearm mastery feat at level 4. That'll give you butt-stroke as a bonus action. With shillelagh, it's doing 1d8 rather than 1d4.
Cast Hex

After a startup of one turn where you only hit twice, you get three attacks per round of 1d8+1d6+3(cha)... which is about on par with a ranger who's doing the same thing, except that your hex spells refresh on a short rest, and you have hexblade's curse 1/day to play with.

Really, though, you're trying to optimize the wrong thing. "Blender optimization" is never going to get you a druid or a cleric. Beyond those, the only class that cares about wisdom is monk, and they can already use the staff with dex. They also don't have any reason to not use it two-handed, which makes it 1d8 damage anyway... so monks have no need for shillelagh. The non-monk potential blenders out there can all use martial weapons one way or the other, which will get you that 1d8 damage pretty readily, which means that the only thing it nets you there is the ability to take polearm mastery and deal 1d8 on the butt-stroke... if you have that feat. Glaive does it better, too, which means that you pretty much also need to be using a shield.

Basically, shillelagh is there to give druids something they can do in melee. It's good at that. It's not good at anything else.

You can do this, but there's really not much there to squeeze.