I would change the name. While "Deathseeker" sounds really cool, I think it would be better as a name for a subclass ability or different subclass that actually seeks death. I personally would change this to "Death Lurker" or "Death Slayer", but if you're tied to the name, that is totally fine. 

This is not meant to criticize you at all, only meant as a tip. In general, unless I have finished balancing, designing, and playtesting a homebrew feat/subclass/race, I do not publish it. Once I have finished it to the extent that I am satisfied with, I publish it. It makes it easier to make new versions of it without having to publish it more, as well as not solidifying anything until I feel comfortable with it. You may want to keep this in mind for the future.

I like the theme, I first expected a subclass similar to a ranger version of the Spores Druid, but that's not what this is. I like that. This is a necromancy-channeling, undead hunting, edgy ranger. 

This seems to be a necromancy based Hunter-inspired subclass. I have a few concerns and recommendations. 

Deathseeker Magic

Inflict Wounds and Animate Dead are good. I would change Death Ward to Blight, Gentle Repose to either Ray of Enfeeblement or [Tooltip Not Found] to either Negative Energy Flood or Enervation.  

Deathseeker's Hand

I would change this name, as the "Hand" doesn't seem to fit that well. I would change that to "Clutch" or "Grasp". 

Devourer of the Living. This is fine, but a bit underpowered in comparison to the Fey Wanderer's feature. If you make this scale to 1d6 damage at level 11, that should be fine. Or, you could change this to 1d8 necrotic damage, only applying to a creature that has taken damage since the start of your last turn. You could even change this to allowing you to choose Radiant or Necrotic damage. 
Igniter of the Fallen. The name is a bit clunky and not great (again, I mean no offense). I would change the name to something like "Punish the Fallen" or "Rebuke the Undead". 
Warden of Balance. This is fine. I can see a lot of Aasimar and Nyxborn choosing this subrace to already have this benefit. 

Lifeweaver

Again, I would change the name a bit. "Weaving" doesn't seem to be connected to the theme of the subclass, and these features aren't so much "life" focused as they are "avoiding death". Also, I don't think these features emulate the theme of the subclass that well, as they are all for avoiding death, while the subclass's fluff text is "They believe that obtaining equilibrium between the living and the undead will usher a new age of peace and harmony, where death is no longer something to be feared." If they don't fear death, why are they trying to prevent it? (This is also my main reason for recommending changing the Death Ward and Raise Dead spells.)

Death Warden. I would change this to the Half-Orc feature that lets you fall to 1 hit point when you would fall to 0, allowing it once a short rest. 
Empowering Vigor. This is way more powerful than the current Death Warden, and is basically the capstone ability of the Life Cleric, but only on yourself. I would make this be once a long rest if you want to keep it. 
Walker of Twilight. I would change the name of this, it just doesn't fit the theme of the subclass. "Death Walker", or something like that should work if you're going to keep this. I personally would get rid of this and merge it with Death Warden, but I honestly don't have any thoughts for how to change it if you don't want to get rid of it. 

Undying Zeal

The first part is fine, but the second part is way too powerful, especially for a ranger. I would change this last part to a feature that does something like this: "Once a long rest, as a reaction, you can reduce the damage you take from one source to zero".

Deathseeker's Might

Bane of the Dead. I think this is fine. 
Blade of Life and Death. This is mostly fine, but I would change this to 2d10 of each damage type and allow you to expend a 3rd-level spell slot to regain a use of this feature. 
Grim Demise. I don't know any DM who has creatures become knocked unconscious as a base, instead of just killing them. This seems weak, especially in comparison to the other two options, but I like the idea. First, get rid of the "kill something you knock unconscious as a reaction," I would change this to "If you deal damage to a creature with a spell or attack, they cannot regain hit points until the end of your next turn", and allow you to create an amount of permanent zombies equal to your Wisdom modifier, which last until killed. 

 Original: 

Deathseeker rangers wander the world, trying to moderate the balance between the forces of life and death. They believe that obtaining equilibrium between the living and the undead will usher a new age of peace and harmony, where death is no longer something to be feared. However, while they want perfect stability between the dead and they living, they won't hesitate to slay an enemy that attacks them, since their own survival is crucial to reaching this balance.

New:

Deathseeker rangers wander the world, trying to moderate the balance between the forces of life, death, and undeath. They believe that obtaining this equilibrium will usher in a new age of peace and harmony, where death is no longer something to be feared. However, while they want perfect stability between the dead, undead, and living, they won't hesitate to slay an enemy that attacks them because their own survival is crucial to reaching this balance.

 granting the ability to speak to the dead somewhere in the class would be really cool, maybe in a feature similar to the monk's tongue of the sun and moon, called "Speech of the Deceased" or something similar.

A feel like the class has a lot of very combat-focused options. I know the base class of the ranger is very exploration oriented, but there are many necromancy/life and death abilities that could be cool socially (Like speaking to only one person, when you are not even in the room, or being able to manifest ghost-like illusions) I think you should combine a few of the existing options and tone them down, adding in a few non-combat options in their place. another way to do this is to just replace one of the features with a non-combat feature instead of three options.

Deathseeker Magic. i really like it! maybe you could switch raise dead to negative energy flood?

Deathseeker’s Hand. i like it, but its a little op.

Devourer of the Living. maybe something like you can only use the feature a number of times equal to your level? or maybe only against your favored foes? its a liile op in its current form

Igniter of the Fallen. this one if fine i think. its powerful, but only against undead, so i think its fair.

Warden of Balance. two resistances at level 3 is very powerful. to balance it, you could make it a choice? or a choice after each long rest?

Lifeweaver. some of the options are kinda op, others not.

Death Warden. this one seems fair to me.

Empowering Vigor. i really like this, but its very powerful. maybe you can only do it a number of time equal to your proficiency bonus or something?

Walker of Twilight. this also seems fair for level 7.

Undying Zeal. a little op, but not a lot. not sure how to fix it. 

Deathseeker's Might. this seems fair. i really like it, especially the grim demise.