This is definitely late, but maybe someone else will be able to come in and take this piece of advice/wisdom I leave.
As for this character looking to achieve an infinite army of Skeletons there is only one choice of class and subclass. And that is the School of Necromancy Wizard. Most for it's 6th and 14th level features.
At 6th level, you add the animate dead spell to your spellbook if it is not there already. When you cast animate dead, you can target one additional corpse or pile of bones, creating another zombie or skeleton, as appropriate.
This feature will help our aspiring Necromancer grow their Skeleton army much faster than any other Necromancer.
Starting at 14th level, you can use magic to bring undead under your control, even those created by other wizards. As an action, you can choose one undead that you can see within 60 feet of you. That creature must make a Charisma saving throw against your wizard spell save DC. If it succeeds, you can’t use this feature on it again. If it fails, it becomes friendly to you and obeys your commands until you use this feature again.
This feature will help us gain an ally who can assist in the creation of more Skeletons.
And for some final preparations were gonna want a few notable spells and even some magic items. Starting with Spells we're gonna need Animate Dead which will be freely given from our subclass. The 7th lv spell Simulacrum. The 9th lv spells Wish and True Polymorph.
With this particular combination of spells we can ensure ourselves a steady steam of Skeletal friends and other nasty undead if we decide our hoard of Skeletons aren't cutting it (or when the Commoners bring out their clubs).
To start we're gonna want to use Simulacrum targeting ourselves (of course) you can use Wish to duplicate the effect of the spell or just spend the time and resources to manually cast it. Now we have a duplicate of our self missing a 7th, 8th, or 9th lv slot (using wish is arguably the better choice). From here you can choose to do an infinite Wizard combo so that you have access an infinite number of yourself that has the Command Undead feature or you can stop at 1. However we move on to the next step which involves making your Simulacrum "Real". This is accomplished by True Polymorphing it into you. As you and the Simulacrum are the same lv. And by virtue of the True Polymorph it becomes permanent if you hold the 1 hour concentration. And now it can regain it's features, spell slots, and hit points in this new form only. (You can do this on yourself to create a cool buffer of hit points and double your spell slots during a day without resting.)
Now we're going to go recruit 2 (or more if you have infinite simulacrum Wizards) new friends who will be helping us build up our army of Skeletons. These friends will be Mummy Lords who are eligible to taken permanently under our control due to their 11 Intelligence. Sad part is they have 11 Intelligence, a +8 Charisma Saving Throw bonus, and Magic Resistance (which is redundant, but worth mentioning). However if you are able to find and enthrall 2 or more Mummy Lords you are sitting pretty with a crew of (at least) 4 Animate Dead users all of whom can recover their spell slots (if you true polymorphed your simulacrum. In a single day if we spend all our slots using Animate Dead which is totaled at 12 3rd lv spell slots, 12 4th lv spell slots, 8 5th lv spell slots, 6 6th lv spell slots, 4 7th lv spell slots, 2 8th lv spell slots, and 2 9th lv spell slots. We create:
18 Skeletons using Animate Dead at 3rd lv.
42 Skeletons using Animate Dead at 4th lv.
56 Skeletons using Animate Dead at 5th lv.
46 Skeletons using Animate Dead at 6th lv.
40 Skeletons using Animate Dead at 7th lv.
24 Skeletons using Animate Dead at 8th lv.
28 Skeletons using Animate Dead at 9th lv.
For a grand total of 212 Skeletons raised in a single day at the cost of 15 minutes. So economical, But that's simply how many we can raise in a single sitting without using a Short Rest or Long Rest. So how many can we maintain if we just simply do the same, but recruit more Skeletons to our cause?
54 Skeletons using Animate Dead at 3rd lv.
78 Skeletons using Animate Dead at 4th lv.
86 Skeletons using Animate Dead at 5th lv.
64 Skeletons using Animate Dead at 6th lv.
52 Skeletons using Animate Dead at 7th lv.
30 Skeletons using Animate Dead at 8th lv.
34 Skeletons using Animate Dead at 9th lv.
Now we're at a daily maintenance of 398 Skeletons a day. The majority of Which have 20 extra hp and each deal 6 more damage when they hit. Now if we do the infinite Wizard combo using true polymorph to make each one "effectively real" you can exponentially increase these results until you're fielding the US Armed Forces in Skeletons (which would be possible if you teach them how to operate such vehicles and equipment.) To be make the math a little simpler. Each copy of yourself creates 98 in a single day using all applicable slots and maintains 143 per day using all applicable slots. Each Mummy Lord creates 29 in a single day using all applicable slots and maintains 56 per day using all applicable slots.
The Mummy lord is our best aid in this regard because they are able to cast Animate Dead and come under our control permanently. However seeing as they aren't a Wizard nor under the School of Necromancy they produce less Skeletons per casting and cap out on castings quicker as they only have 9 spell slots that can be used to cast Animate Dead. While our True Polymorph Simulacrums are Skeleton making Machines.
Now theoretically you could create a Simulacrum get them a Mummy Lord then True Polymorph them into yourself enabling them to have a second copy of the Command Undead feature that can get each Simulacrum 2 Mummy Lords. The only RAW thing I see stopping this from happening is the Command Undead feature itself.
If it fails, it becomes friendly to you and obeys your commands until you use this feature again.
This line might refer to the creature itself using the feature a second time at all. Or it could be referring to that specific instance of the feature.
Anyways from there you could do it yourself and get two Mummy Lords doubling the Skeleton production for your Mummy Lords. And at this point you could save your own spell slots for being the Utility Wizard you are and instead allow your Army of Simulacrums and Mummy Lords to take up the task of creating and maintaining your army of Skeletons. You are now the Commander in Chief of the greatest Skeleton Army and nothing short of a Deity or the DM can stop you.
So with the scenario I have presented you can exert influence over an infinite number Skeletons while also having the added the benefit of being a Wizard and having a likewise infinite number of Simulacrums and Mummy Lords.