Short Answer: No. Not in 5e, yet anyway.
I checked the Abyss book hoping there was something, but there wasn't. In previous editions, there were several ways to do it. BECMI Immortal rules, Necromancer's Handbook, and Planescape, for instance.
HOW TO DO IT LEGALLY
So from those, here's some Setting As Written and Rules As Written approaches, to getting you toward a Fiendish Template without a Polymorph spell changing your race.
First, you will need to change your home plane to a lower plane, instead of the prime material plane. How you do this varies. You can do this and just about everything else by using a wish, but if you can find an easier way, you should.
Second, you want to have a way of getting to your plane. That will help represent your nature better. There is a Boon here:
Boon of PLANAR TRAVEL
When you gain this boon, choose a plane of
existence other than the Material Plane. You can now use an action to
cast the plane shift spell (no spell slot or components required),
targeting yourself only, and travel to the chosen plane, or from that
plane back to the Material Plane. Once you use this boon, you can't
use it again until you finish a short rest (DMG. p 232)
Third, you want to live as long as lower planar creatures live, which is irritatingly enough, forever.
BOON OF IMMORTALITY You stop aging. You are immune to any effect that
would age you, and you can't die from old age.
Next, you will want to be able to survive the total terrain of, well, you know, lakes of fire, and all those breath weapons and fire balls being used by your neighbors.
BOON OF THE FIRE SOUL You have immunity to fire damage. You can also cast burning hands (save DC 15) at will, without using a speL
slot or any components.
So far, you still look like you, act like you, but you now live in hell, will live there forever, and can swim in the lake of fire with your neighbors. If you are killed on the Prime Material Plane, in theory, if you changed your home plane, you should be banished to that Lower Plane you switched to. This also means your Prime Material Type has been changed to a Planar Material type, making you subject to all the restrictions, banishment, wards, etc.
These rules do not currently exist in 5th edition in a direct way. They are legally touched upon in the individual spells, but unlike editions with Planescape, you will have a lot of work to do to compile all the information and rules as they apply to the 5th edition. Like I said, a wish probably has about a 100% chance of being able to do this, but how your DM rules the wish, or whether they want it to succeed makes it safer to try other approaches, rather than being default polymorphed into some low CR fiend like a lemure.
The boon of perfect health would probably help, but not all fiends are immune to things like all diseases or all poisons, only some of them, on average. Technically, there's a few fiends not immune to fire - the ones on layers of ice, so its not a necessary condition either. Only changing your home plane is really critical, and since its a plane of dead petitioners turned into immortal demons or devils, you too will have to be immortal just to keep up.
Speculation, not RAW
Take the Actor Feat if your Charisma isn't maxed, (i.e.e liar/charlatan/disguises and a forked tongue charisma bonus), and then get really good at summoning and leading fiends. There used to be a blood war in the lower planes. If your DM still uses it, sign up and lead a small army or group against high value targets. Bonus points if you trick good PCs into helping you. This builds up the reputation you need among the NPC demons and devils.
Certain Eldritch Invocations from your warlock class can also help build the myth. Like Devil's Sight, Mask of Many Faces/Myriad Forms (the higher ranked devils tend to assume many different forms to deceive people), and one with the shadows to disappear anywhere near darkness.
As a non mechanical DM approach, you may attempt to successfully take out the competition or one of your Archfiend's Rivals directly, and in a humiliating way, such as through a successful Imprisonment spell. These escapades lay the ground work for qualifying for Boons, like the ones above.
True Names & Promotion
Promotion and Demotion. When the soul of an evil mortal sinks into the
Nine Hells, it takes on the physical form of a wretched lemure.
Archdevils and greater devils have the power to promote lemures to
lesser devils. Archdevils can promote lesser devils to greater devils,
and Asmodeus alone can promote a greater devil to archdevil status.
This diabolic promotion invokes a brief, painful transformation, with
the devil's memories passing intact from one form to the next.(MM p.67)
DEVIL TRUE NAMES AND TALISMANS A mortal who learns a devil's true name
can use powerful summoning magic to call the devil from the Nine Hells
and bind it into service... However it is summoned, a devil brought to
the Material Plane typically resents being pressed into service.
However, the devil seizes every opportunity to corrupt its summoner so
that the summoner's soul ends up in the Nine Hells. (MM p.67)
You can see that the rules as written are really keen on turning you into a low level nearly mindless weak demon, following the traditional promotion path. The only real way to circumnavigate that is to have some kind of magically binding contract or artifact, or literally take over the throne of an existing quasi-power, for example, by using the above imprisonment spell and successfully using Deception, Intimidation, and Persuasion to take over their lieutenants. But this is mere speculation of a Hostile Takeover.
Unfortunately, 5th edition still lacks considerable amounts of useful info on player characters and the planes.