I have a deep affection for the brooding, depressing, paranoid world of the spy thriller. While I do enjoy an action-packed James Bond movie, the more realistic and pessimistic version of the genre, particularly from the works of John Le Carré (who actually worked in British Intelligence, and had to get special permission from his bosses to publish The Spy Who Came In From the Cold while still working in that field) inspires me more.
In a lot of ways, you get the grimy back-rooms and shady characters of a Noir-ish crime drama, but with much higher stakes.
Now, D&D is a game about fighting monsters, and so this is not a build that would necessarily work in a one-size-fits-all party. Instead, this would have to be for a campaign that really focused on spycraft and espionage.
When you think of the ultimate spy character in D&D, you probably think of a Rogue. I actually think you'll be a bit disappointed if that's the route you go. Rogues (setting aside their shortcomings as martial warriors) are built to move swiftly and silently, to be assassins and thieves (I mean, those are literally two of their subclasses).
But the way actual Cold War spycraft worked wasn't really how Hollywood portrays it. It wasn't the American CIA officer who snuck into some Soviet research facility to take photos of their new submarine plans. Instead, the officer would recruit someone working at that facility, meet with them in secret, and persuade them to take the extremely high risk in exchange for something - usually money, and perhaps a promise to eventually exfiltrate them to start a new life in the US.
And that's why I think that the ultimate spy in D&D would be an Eloquence Bard.
I'll note here that I think Liam O'Brien had a similar idea in Campaign Four of Critical Role. While his character, Halandil Fang, is more like the William Shakespeare of the setting (a brilliant playwright and head of the city's most prominent theater company,) the tasks set to him at the "Schemer's Table" have been very high-stakes spycraft (a little more of an underground revolution than Cold War spying, but similar "maintain a plausible cover while secretly undermining the powers that be" drama).
First and foremost, the thing that the Eloquence Bard gets right from level 3 is Silver Tongue, which is effectively the Rogue's Reliable Talent, but specifically for Persuasion and Deception checks. Admittedly, at low level, getting no less than a 10 plus your modifier on these rolls might still not cut it in really dire situations. But if we grab Expertise in these skills, and assume we start off with a +3 to Charisma, we'll be rolling no lower than a 17 by level 3.
These skills are going to be really core to what we do as a spy. Persuasion can sometimes be used as a "misleading the target by telling the truth in a certain way," but it's also going to be crucial if you want to persuade assets you meet to do what you need them to do.
Now, I'll allow for the following: you might actually want to start off at 1st level with Rogue, because you will get one more skill proficiency this way. Multiclassing with both will allow you to get more expertise, though I think it's possible we'll want to just stick to Bard for a little bit for spell progression.
Other skills that I think will be very helpful to have are Insight, Perception, Stealth, and possibly Investigation. As a Bard, we're not going to get all of those, but we could pick a species that gets one of them as a bonus (Elves, for instance, can get Perception for free - and the fact that they never technically sleep and can actually keep watch while Trancing could be very helpful).
We'll naturally want to focus on Charisma as our best stat, but I'd say after taking care of enough Con to have some meat on our bones, we probably want to have a bit of Wisdom. It is tough - as a Light Armor class that doesn't actually focus on Dexterity, we'll probably have a terrible AC, so you might instead push Dex, which will help with Stealth.
I'll also say this: there's a case to be made for grabbing a bit of Sorcerer here - primarily for Metamagic, as we'll likely want to have Subtle Spell available to us.
One of the biggest challenges that an intelligence officer has in a hostile place is that their movements are being tracked constantly. It was crucial, of course, to not actually be identified as an intelligence officer, because at best you'd have your diplomatic credentials revoked and be deported, and at worse, they'd capture you and interrogate you (probably under torture,) and maybe kill you (though as I understand it, it was the cultivated assets who tended to be tortured to death).
Thus, any means to be able to do your work without making it clear you're doing your work can be helpful. Still, depending on how your DM rules it, you might be able to get some spells off without Subtle Spell and still go undetected if you, say, break line of sight and maybe are allowed to mutter the verbal components under your breath.
So, what spells might we consider taking?
For cantrips, I think Message feels pretty obvious - if you could arrange for your asset to sit in a public place and do something innocuous (akin to reading a newspaper) while you're in an inn looking down from a window on an upper floor and can just message into their heads, you could get a lot of discussion done without the asset seeming to be speaking at all.
Prestidigitation actually could be really great - if you need to make a Dead Drop (where you don't actually see your asset, but just leave instructions or equipment for them stashed away somewhere) you could identify the place it's hidden with a secret mark that then, conveniently, disappears after an hour.
Next, for 1st level spells, Charm Person might actually be best used when cultivating an asset - you charm them and then make your case. They will know that you charmed them when the spell ends, but you might find a way to put them into a position where they'll forgive you.
Disguise Self is a great option, though I'll throw this out there: you might instead want to get proficiency with a Disguise Kit. Disguise Self is thwarted by Dispel Magic, or even Detect Magic, and Truesight. A mundane disguise does not set off any pings in any kind of magical detection. I don't think it would even fail under Truesight given that it's not like this isn't your "true form."
Illusory Script - yes, it's tough to justify this on a Bard, as compared to a Wizard - is tailor-made for this kind of thing. You leave a dead drop instruction for your asset, and if someone happens to stumble upon it, the writing might be something innocuous.
Naturally, any kind of illusion spell, from Minor Image to Silent Image to Major Image to Programmed Illusion, could be very useful in myriad ways.
Onto second level spells:
You might consider grabbing Enhance Ability both for your party but also for your assets. The duration isn't super-long, but if you can make contact before they go on a dangerous endeavor, it will make them more likely to succeed.
Detect Thoughts is also pretty obvious: both for finding hidden listeners and also getting better angles to work on people.
Invisibility is obviously pretty helpful.
Suggestion is a pretty powerful one - you can effectively make an asset of someone if they fail a wisdom saving throw. While this probably won't be as reliable as someone you really coach through things, it can work in a pinch.
Now, rather than going through every spell level, I'm going to just talk about a few highlights. Naturally, as a Bard, at level 10 we can start getting any spell from the Wizard, Cleric, or Druid list, so the possibilities become very broad.
Nondetection could be helpful, but I think if you have to cast this constantly to avoid being detected, you're probably screwed. It might work if you're sending someone in where they're not supposed to be, but the whole point of cultivating an asset is that they are supposed to be there. Now, it could help your asset avoid getting their mind read via Detect Thoughts, but that's a pretty specific situation.
Mislead, a 5th level spell, is basically built for spycraft. You simultaneously go invisible while an illusory double of you takes your place. It's kind of what Loki does in the Thor movies, and if you're being tailed, you can send the illusion around a corner and have it vanish, or even have it just sit at a cafe doing nothing for an hour while you put some distance between you and the people surveilling you.
Modify Memory feels like a real must-have, the kind of major ripcord you might need to pull if a foe witnesses you.
Rary's Telepathic Bond can give you a really lasting connection with your asset in case they need guidance while they're on a mission for you.
Mind Blank and Glibness can both be really helpful. Frankly, as someone running a campaign with a high-level Bard, I kind of hate glibness because it just says "I succeed on all Charisma checks" (he's got expertise in all four charisma skills,) so when his character needs to do a high-stakes argument, there's very little interactivity to it.
But Mind Blank is one that could really help you if you're coming up against enemy mind-readers.
I will say that I don't know how sustainable a grim Cold War spy campaign will be at higher levels. I think Dark and Gritty in D&D really works best in tier 1, where bad luck or bad decisions can truly put you in a bind. The higher-level players get, the more ripcords they have to pull, and while you might be able to sustain the tension by finding some conceit for you to still need to empower low-CR NPCs to do these tasks, at a certain point, the game really allows a high-level character to have a better chance than any Apprentice Mage or Warrior Veteran.
Still, I think you could have a lot of fun if you played a character like a real spy, if a DM is willing to buy in to that fantasy.