Ultra Mutant is the rarest achievement in the game, and while offering varying levels of difficulty a big chunk of this is just a long grind. It requires all 12 characters to be unlocked and all 12 B-Skins to be unlocked as well(which requires a Golden Weapon on each character for YV's B-Skin). The most daunting part is it requires all 12 crowns unlocked on each character - 72 preloop unlocks and 72 loop unlocks. The entirety of the Tech Page covers mechanics for every single character and crown in-depth, so if any character or crown seems confusing there are strategy guides on each to help with this struggle.
What does not count towards Ultra Mutant is secret character unlocks. Crowns can technically be unlocked on secret chars but cannot be accessed nor does it effect anything. Hardmode must be unlocked in order to get Chicken's B-Skin, but Golden Disc/Nuke is not required. Any golden weapon unlock will do for each character.
Here is a helpful google sheet tracker usable here. This is a great way to keep track of crowns as there is no way to see what is already unlocked mid-run. (Also, gives a better sense of completion!)
B-Skins & Golden Weapons
Many B-Skins are unlocked before the Ultra Mutant grind even begins, and the rest can just be gone for during the crown grind. There are a few exceptions to this case;
Melting's B-Skin is a particularly hard B-Skin challenge, and as such there is an entire guide dedicated to just that located here.
Chicken's B-Skin can be annoying to those unfamiliar to hardmode, but the task isn't actually too demanding. One particular useful strategy is to go Golden Grenade Launcher + CoDeath to help kill elites more easily.
YV's B-Skin can come naturally while playing as golden weapons frequently appear in loop. However if rng is too annoying then one can store a golden weapon in the proto chest in the first vault (requirement there is to be carrying any crown) as any character with these unlocks, and then re-visit the vault as a character without this unlock.
The most efficient use of time to cut down this achievement grind is to go for a crown unlock on any run that a B-Skin is attempted. Even if not going for Ultra Mutant yet, it never hurts to stay a step ahead!
The Crowns
Now into the meat of the matter - the 144 crown unlocks. Being able to get this many unlocks requires a near mastery of every character and their mechanics, so being familiar with each character's mechanics goes a long way. With 72 preloop and 72 loop crowns, the most efficient time-wise is to unlock a crown in preloop, and then unlock a loop crown as well. This can potentially cut the number of runs needed in half.
On the current steam versions going with the default(u99r1) is recommended. openbeta from 2021 may not show unlock messages as a bug but should work. 2023 beta(win_64) currently crashes on unlocks and should be avoided.
Dailies and Weeklies both allow for crown unlocks. This is particularly useful as there is a community run sheet scouting sets. Seeing ahead potential routes can help plan for what crown unlocks to go for on that specific day. Loop crowns can be unlocked in preloop if there is a Weekly that features them, and suggesting/voting weeklies is a feature on the Nuclear Throne discord. There are some people who vote just based on what unlocks they're going for at the time. (It should be noted that golden weapons specifically cannot be unlocked on weeklies only.)
Preloop Crowns
Within preloop there are two vaults, 3-2 and 5-2. These can be missed by going into YV's Mansion / Jungle secret areas respectively, so make sure not to accidentally miss out. While beating preloop is particularly easy at this point in the game, the crowns have varying effects that can make the game unnecessarily difficult because of their downsides. Because of that, its better to take some crowns only on 5-2 as only 5 levels need to be beaten with that crown in order to unlock it. (An additional 3 areas if going for loop vault to do a double unlock).
In order to unlock crowns in preloop one must either sit on the Nuclear Throne by killing it, or loop by destroying all 4 gens and then killing the Throne, entering the loop portal.
Any run with Haste/Curses and probably Destiny should be runs where double unlocks are gone for. The other crowns can be more situational that can make the first 3 levels in loop particularly challenging, but if the option is there then may as well go for it to try to save time.
Loop Crowns
Within loop there are 4 vaults. On top of 3-2 and 5-2, one spawns on 1-2 and 7-2 as well. However, once three vaults are visited vaults will no longer spawn in the game. There are three ways to unlock crowns in loop; sitting on the loop Throne, looping past the Throne, or a new choice where Captain can be defeated for the secret ending which also unlocks crowns. The two most popular strategies is to take a crown from the 1-2 L1 vault, immediately enter HQ on 1-3 L1 and defeat Captain; the other being to avoid visiting vaults until 7-2 L1 and needing to defeat only the Throne with that specific modifier. These both can have their upsides and downsides, but from the perspective of this guide Captain is far easier once her attacks are understood and countered.
While no crown is offensively bad unlike preloop, some of them are just pretty forgettable. However this means no one crown should be particularly challenging, with the exception of Crown of Blood which can be daunting on the wrong set.
Preloop has a total of 15 normal levels that must be beaten with 4 bosses, while loop has a total of 16 levels(includes campfire) with a total of 8 bosses that need to be defeated. So going to the end of Loop 1 to unlock a crown can be seen as risky, the Captain strat has attacks that make her one of the most dangerous bosses if not knowing how to properly deal with her. However, the HQ route if done from 1-3 L1 only requires 6.5 levels to be beaten, though 2 of those may be filled to the brim with elite IDPD.
(Write HQ section another time, after strategy guide)