No, it would be a normal skeleton. True Polymorph is not a necromancy spell, so Undead Trall's benefit would not happen. It would apply to the zombie created by the initial Finger of Death, but True Polymorph would wipe out that benefit when it changed the zombie into a standard skeleton. (This is, of course, assuming the increased HP and damage didn't raise the zombie's CR and break the combo).
I've always seen it ruled that Finger of Death dropping you to 0 hp would turn you into a zombie. However, a recent Q&A tweet by Jeremy Crawford and a subsequent re-reading of the spell has clarified how this spell works.
Question (@DanDanFielding)
For Finger of Death: if the spell brings the target to 0hp and the target subsequently dies due to failed death saves, dies that count as FoD killing the target? Will it rise as a zombie?
Answer (@JeremyECrawford)
A spell kills you if its damage or other effects slay you. If it reduces to you 0 hit points but leaves you alive, it didn't kill you. #DnD
Let's take a closer look at how dropping to 0 hitpoints works in 5e:
Dropping to 0 hitpoints: When you drop to 0 hit points, you either die outright or fall unconscious [...]
The rules go on to explain that dropping to zero hitpoints knocks you unconscious and causes you to start making death saving throws. Dropping to zero only kills you outright if the remaining damage exceeds your maximum hitpoint total. For example:
[...] a cleric with a maximum of 12 hit points currently has 6 hit points. If she takes 18 damage from an attack, she is reduced to 0 hit points, but 12 damage remains. Because the remaining damage equals her hit point maximum, the cleric dies.
To apply this to our example, let's say a player has a maximum of 30 hit points and currently has 20 hit points left. If that player is hit by a Finger of Death for 62 damage, they'll be dropped to 0 hit points with 42 damage remaining. Since that damage exceeds their hit point total, they will die outright and rise on the caster's next turn as a zombie under the caster's control.
Another common scenario that could lead to Finger of Death turning a player into a zombie would be if that player has already failed two death saving throws before the spell hits them. When they take damage from the spell, that would count as failing a third saving throw, which would kill them outright and raise them as a zombie on the caster's next turn.
However, if the player hit by the spell is not currently dying, and is dropped to 0 hit points without dying outright, the spell does not raise them as a zombie if or when they die. It simply does a very large amount of necrotic damage.
Best Answer
Yes, it is possible
It is actually possible to obtain a permanent skeleton companion. That being said, it is a slightly convoluted process that is not without certain shortcomings. This method involves using two spells in conjunction:
Finger of Death
True Polymorph
By now, you probably understand where I'm going with this. You would first start by using Finger of Death to create a normal zombie servant under your control. Next, you would use the True Polymorph spell to transform said zombie into a skeleton (which you can do because they are both challenge rating 1/4). Then you simply wait out the spell's duration, which makes the transformation permanent, and, since a transformed creature "retains its alignment and personality" you now have one permanent skeleton companion at your disposal.
In fact, using this method, you could create a permanent companion out of any creature of 1/4 challenge rating you may desire.
In Summary
Finger of Death + True Polymorph = permanent skeleton companion