UPDATE: I've finally unlocked this achievement after over a year of grinding. This achievement will take a very long to unlock.
So you really want to get this achievement? For real? Well then, prepare for a very, very long grind towards getting this.
To unlock this, you have to earn a total of 50000 reknown across all of your underworld factions. For many, this may be way too much to attempt but if you really want to go for this, be prepared to grind this out for months on end. You can now buy new delve attempts for 50 gems, but costs increase with each subsequent purchase so a second purchase will be 100 gems, a third being 150 gems and so on. The cost resets to 50 on the next day.
GET YOUR FACTION REKNOWN AS HIGH AS POSSIBLE!
To start, you have to finish as many factions as you can by reaching 2500 reknown or at least getting all faction reknown as high as you can handle on each faction. Some factions are incredibly difficult or luck based and may not be worth attempting until the treasure hoards are 225 or higher (unlocks power potions boosting stats by 7). Others are reasonably easy and you can finish without getting treasure hoards anywhere that high. You don't actually have to get 2500 on all factions but you must get their reknown as high as you can possibly get them and have as few factions below 1000 reknown as possible.
It is best to work on easier factions such as The Deep Hive and then the medium difficulty factions such as Hall of Guardians and Illithia, and avoid the hardest ones (Hell Gate, The Labyrinth, Lyrazsa's Lair) until later on. You should aim for getting as many factions to 1300 reknown as possible and limit having any below 1000 as much as possible.
For reaching and beating Level 500, I used two separate runs at this when completing a faction. One technique that worked for me was to level up my treasure hoard on a respective faction in order to beat level 500 with pure faction teams. You can use duplicates of a faction's troop as long as all four slots are occupied by troops from that faction (I.E. Three Lunas from The Warrens, or Two Chief Dargons on Fang Moor). This method is not affected if one or more of the troops dies. However it will negatively affect the 'no casualties' run which is why I recommend a separate run afterwards, depending on your preference of order.
On the 'no casualties' run, I used three faction troops and either a Cauldron (doubles attack and adds life) and making the first and second troops extremely lethal in terms of dealing skull damage or a Celestial Flask for the chance of creating mana potions in order to create gems for certain troops that use red, green, yellow, and purple mana in order to fill up their mana in order to cast spells on enemies. The former is excellent for getting through difficult factions that may be impossible to complete with pure factions early on and the latter is excellent for using troops that cast splash damage. I used the Cauldron when working on beating level 500 and hopefully avoid any casualties and did the pure faction separately. This won't always work right away. Either the treasure hoard is not high enough to boost stats enough or the board simply screws you over by generating one bad lineup after another. Be ready for those aggravating and even infuriating moments where a bad lineup of gems/skulls ruins the entire delve. I've had situations like this force me to abandon delves before even getting past room on even with treasure hoards at 225 or higher.
You can also simply level up the factions' treasure hoards to boost the teams' stats on a respective faction and with potions added, they help A LOT against high-level enemy delve teams. These potions are unlocked by reaching treasure hoard level 100 (Enchantment), 105 (Explosions), 115 (Rejuvination), 130 (Armor), 150 (Blessing), 175 (Shielding), and finally 200 for the Potion of Power which adds 7 to shielding, health, attack, and magic to all of your troops . Every 25 levels beyond hoard level 200 adds another power potions so if you're struggling badly, you'll simply have to keep adding treasures to reach hoard level 225, 250, and beyond to have a shot at finishing some factions. This is very expensive to do since it costs LOTs of gold to do so be careful if you're working on guild tasks. Completing the factions can also help deal with 'pet blocked' kingdoms by unlocking the factions respective pet.
You will unlock "Kind of a Big Deal" for 1000 reknown, "People Know Me" for 5000 reknown, "I'm Very Important" for 10000 reknown, "I have Many Leatherbound Books" for 20000 reknown, "My Kingdom Smells Like Rich Mahogony" for 30000 reknown, and "Well I'm Very Happy For You" by reaching 40000 reknown. If you reach 2500 on a given faction, you will also unlock "Exalted" as well.
This achievement is not easy nor for the faint of hearted gamers. It will take an incredible amount of work on multiple factions and in-game resources to complete, even if you're a high-level player like I am (I'm currently level 1267 as of this writing).
Of course, it's just a matter of time before the developers release an achievement for 60000 reknown and beyond! Just be prepared for a very long grind if you decide to go for this.