Balance
So firstly, you need to understand that you're effectively stealing a class feature. Wizards get the least class features of any class, because their access to a wide variety of spells makes up for it.
With that said, balance-wise, this is not going to be an issue. Necromancy spells come in 3 broad categories: Offensive, resurrection, and minions. The offensive spells don't do anything that you can't do anyway, so no problems there.
The resurrection spells are incredibly powerful, but it doesn't matter. In most campaigns, a dead character will either be resurrected or replaced in short order, so this is effectively just making it easier for the GM.
The minion spells are interesting, because rolling around with a horde of zombies is quite powerful - for a Necromancer. For you, it's going to be pretty weak. Your undead minions won't have the increased health and damage of the Necromancer's, and you'll have half as many as a Necromancer does anyway.
Flavour
Flavour-wise, this is an issue. For a start, making undead is evil. See here for an explanation. So if you want zombies, you're committing yourself to playing an evil character. Of course, this isn't that hard to work with, especially since the GM is clearly on your side here.
The next point is a bit trickier - Druids are followers of nature. And in nature, when you die, that's it. The resurrection-style spells are definitely not the sort of thing Druids are a fan of. The undead, on the other hand, are basically a perversion of everything Druids hold dear. They're a mockery of life.
So with that said, you're clearly looking at playing an evil Druid. Almost an anti-druid. Which sounds really cool! And the DMG (pages 96 and 97) includes evil villain classes. One is the evil Death cleric, which is already pretty close to what you're after, and the other is the oathbreaker, or antipaladin, which is effectively an archetype of a class that is the opposite of the class. So there's precedent for an anti-druid.
Mechanics
You're looking at playing a Circle of the Land druid, which is good, because you can create a custom Circle to give you the spells you want without changing the core Druid at all. And you can call it something cool like Circle of the Grave, or Circle of the Tomb! Cleric domain spells work exactly the same as Druid circle spells, except that Cleric domains include 1st-level domain spells. So one option would be to take the Death domain spells directly, excluding the 1st-level ones. This would give you:
3rd level: blindness/deafness, ray of enfeeblement
5th level: animate dead, vampiric touch
7th level: blight, death ward
9th level: antilife shell, cloudkill
Alternatively, you can make your own. Pick 2 death-related spells at each spell level from 2nd to 5th and you're good to go! So it might look like this (I've tried to pick out some of the better spells):
3rd level: blindness/deafness, ray of enfeeblement
5th level: animate dead, revivify
7th level: blight, death ward
9th level: antilife shell, raise dead
As far as the higher level spells you want are concerned (finger of death, circle of death, power word kill), you might just have to ask your GM to add them to the Druid spell list for you. It shouldn't make a huge difference, but it is worth pointing out that these are extremely powerful offensive spells, which is something Druids don't normally get a lot of, particularly single-target spells.