Short Answer:
If nothing else, play a Human Fighter (Champion) with the Noble background and a one-level dip into Rogue (taken as your second level overall). Put your highest ability scores into Charisma, Wisdom, and either Strength or Dexterity (in that order; dump-stat whichever of Str or Dex you didn't put third). Take the Skilled Feat as your variant human feat. Your skills by Level 2 should include Deception, History, Insight, Intimidation, Persuasion. Apply Expertise to Insight and Persuasion. You are now basically a real-world medieval aristocrat with exceptional diplomatic training.
From Noble: History, Insight*, position of privilege
From Skilled: Persuasion*, (any skill), (any skill)
From Fighter 1: Intimidation, Perception
From Rogue 1: Deception, expertise
*Expertise from Rogue 1
There. Two levels, and you're as badass a diplomat as you're gonna get. You are also not useless in a fight, which is good.
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Longwinded Answer:
There's already a Diplomat class in 5e, and it's called A Single Level Dip in Rogue.
Seriously. Hear me out.
Rogue and Bard are probably the best classes for diplomats, thanks to Expertise (which doubles your proficiency on two skills of your choice). But Expertise is granted by Rogue 1, so it only takes a one-level dip to get it for Persuasion and Insight. Beyond that, all twelve classes can make fine diplomats as long as you give them a race/background/class combo that in some way allows them access to Insight and Persuasion as core requisites, plus Deception, Intimidation, and Perception for support (note that a one-level dip into Rogue also gives you an extra skill from a list that includes all of those). If you want, you can just make a Diplomat background that comes with Insight and Persuasion and one of the countless appropriate background powers from the PHB (it won't break anything). As long as you have the core skills with Expertise, you will be a heroically badass diplomat. All of the other combat and exploration powers you get – and again, you have twelve classes worth of options – are for flavor. Pick whatever you would have played otherwise.
What's really important about this is that it's a very good thing that you can't do better.
Let's imagine for a minute that there's a Diplomat class (or Noble, or Aristocrat, or whatever). It comes with sucky weapon proficiencies, sucky armor proficiencies, no magic, and no combat-oriented abilities. It's just pure diplomacy, all the time.
What this means is that all diplomats will be a member of this class. Because it's stupid to send anyone else. Which means you don't get warrior-kings and wizards and master spies hammering out treaties, you get Fantasy Eurocrats. They're massively overpowered at exactly one thing, so overpowered that there's no point in trying to compete with them at it if you're not one, and they suck at all the other things. Which means the guy playing them isn't having fun unless there's a negotiation, and nobody else is having fun when there is one.
If you've ever played a Shadowrun Face, you know how terrible this is. In Shadowrun, the Face always takes some sort of combat skill. It's just an unwritten rule of the game. Why? Because it's not fun for the other players otherwise. And it's probably not that fun for you, either.
The great thing about 5e is that class is about your adventuring abilities, not your vocational skills. Want to be a sailor? Fine, take the sailor background and then pick any class you want. You don't need a sailor class that gives you a bonus to balance and climb checks and grants special moves with a belaying pin. You can be a Fighter sailor, a Barbarian sailor, a Wizard sailor, or a Cleric sailor. And you're just as good at sailing as any sailor NPC.
So if you want to be a diplomat, just have a decent Charisma and Wisdom and take Insight and Persuasion as skills. If you want to be an exceptional diplomat, better than any NPC of the same proficiency bonus, then take a one-level dip into Rogue for Expertise and an extra skill as well. You can still be any class you want on top of that (though classes like Bard, Cleric, Sorcerer, and Warlock will make better use of the high Charisma and Wisdom than, say, Wizard or Barbarian do).