I want to create a character that can wear the skin of his victims to infiltrate their ranks. Optimally this would be a robot - a warforged-type character. Is there a way I can go about this using pre-existing D&D rules? I'd like to avoid homebrew, but if there are easy ways of doing that, I'm open to it.
Outside of mechanics, talk to your DM/party about this
As others have noted, there are mechanical ways to fluff existing options to this sort of skinwalking, but I wanted to point out another issue: table viability of the strategy.
Whether or not this is do-able is almost secondary to this issue,. I would heavily recommend that you discuss your plans with your DM and table-mates first before going down this path.
Unless you are all playing in a setting or group that is okay with it, this wearing the skins of the deceased may not fly with your DM or with the other players. It's a pretty bold strategy and definitely says something about your character. Unless everyone is on board with this, it may result in issues at the table.
I don't see there being a problem here, but you might want to focus your ability points on two areas: Intelligence (disguising) and Charisma (deception, performance, persuasion). The PHB p. 178 states how Intelligence is the ability score that would allow you to "put together a disguise kit to pass as a city guard". So you want to have a high Intelligence bonus for these checks.
You also might want to have proficiency with the Disguise Kit - which in your case would be a bespoke version of this, i.e. it would include skins. The Entertainer and Urchin backgrounds give you proficiency with a Disguise Kit (PHB pp.130,141). In XGtE p.81 it sets the difficulty for copying a humanoid's appearance as DC 20, so you want to boost your chances on your check as much as possible. If you want to have Expertise (double proficiency) with the kit, you can achieve this, too. See: Is there a way of having Expertise with the Poisoner's Kit?
As a class, I would suggest Rogue because you get Expertise (PHB p.96), and you can double the proficiency bonus on a skill, including Deception. In practice, you get x2 proficiency bonus for your Intelligence checks when you disguise yourself. This is going to be crucial to the purpose you intend. Also, if you get "into a pickle", you are more likely to be able to make it out alive is you have Cunning Action (Dash) and can Hide.
To enhance further, if you find a way of acquiring Enhance Ability, it means you can get advantage on your Intelligence/Charisma checks. This is a 2nd-level spell normally available to a Bard, Cleric, Druid, or Sorcerer. Alternatively, the spell Enhance Ability is available to Wizards in UA Class Feature Variants, so you could choose Arcane Trickster. This would open up useful illusion spells to you as well. If this is an option, that would be my choice.
The other roguish Archetype I would choose is Assassin because of the Infiltration Expertise and Impostor features which you get on reaching 9th and 13th level. This seems to fit in nicely with what you intend. Lastly, Mastermind provides an interesting skill, too, because you get Master of Intrigue at 3rd level which gives you proficiency with disguise and forgery kits -in case you didn't have it already-, two languages, and "you can unerringly mimic the speech patterns and accent of a creature [...] enabling you to pass yourself off as a native speaker of a particular land" (XGtE, p.46).
If you choose Bard as your class instead, you also get Expertise when you reach 3rd level, which is when you would have access to this spell, too.
The Actor feat (PHB p. 165) would be very useful as well as it provides you with advantage on Charisma (Deception and Performance) checks "when trying to pass yourself off as a different person". This, in combination with the Master of Intrigue feature from the Mastermind archetype, will be a powerful deception tool.
On an interesting side-note, if the PC was a follower of Xipetotec the aztec flayed god, wearing another's skin as a disguise may even be encouraged! Your DM can create their own "flayed god" for the purpose of your campaign setting, too.
Yes, an Artificer can do this.
Artificers use artisan's tools that they’re proficient with to cast spells. If you’re an Artificer with proficiency in Leatherworker’s Tools, you could use them to cast spells like Disguise Self or Alter Self (both of which are on the Artificer spell list).
To quote the relevant section of Eberron: Rising from the Last War:
You produce your artificer spell effects through your tools. You must have a spellcasting focus-specifically thieves' tools or some kind of artisan's tool-in hand when you cast any spell with this Spellcasting feature. You must be proficient with the tool to use it in this way. See chapter 5, "Equipment," in the Player's Handbook for descriptions of these tools.
Leatherworking tools are a form of artisan's tools, so they're a viable option to use to cast spells with - and wearing a skin that you'd used them on would be a viable method of doing so (like using temporary potions you'd brewed with Alchemist's Tools would be). From the sidebar Magic of Artifice in that book:
As an artificer, you use tools when you cast your spells. When describing your spellcasting, think about how you're using a tool to perform the spell effect. If you cast cure wounds using alchemist's supplies, you could be quickly producing a salve. If you cast it using tinker's tools, you might have a miniature mechanical spider that binds wounds. When you cast poison spray, you could fling foul chemicals or use a wand that spits venom. The effect of the spell is the same as for a spellcaster of any other class, but your method of spellcasting is special.
Reflavour disguise self
You make yourself--including your clothing, armor, weapons, and other belongings on your person--look different until the spell ends or until you use your action to dismiss it.
You can simply reflavour it so that when you cast the spell, you just put on the skin. It will function mechanically the same.