Consider having the characters designed so that there is some meaningful reason for them to be the ones selected.
I remember one campaign I played (Planescape) in where some red dragon asked us first-level scrubs to go find her precious stolen dragon egg. My response to that was "You must not want it back very much!"
Contrast another campaign (Eberron), where one of the PCs was of the dragonmarked noble house, and the rest of us were retainers or otherwise friendly/bound to him. He gets sent not because he's "high level" (metagame concern) but because he's an important guy, and we get sent because we're his posse. That's a lot better story.
In other words, if these organizations would hire "professional agents" - why are they not generating characters that are professional agents? Or are in that nation's military? Et cetera? The whole "let people generate anything they want and then it's the GM's job to somehow shoehorn them into the party and the campaign" approach is old and busted; don't do it. See this previous question on forming parties for relevant advice.
Of course suitability doesn't have to be "best for the job," it can mean "disposable," "the only volunteers," "some power bloc wants them to go," etc. In the Pathfinder Society, the organized play society for Pathfinder, all PCs have to pick a faction that is then part of the motivating force (and that works behind the scenes to get its agents included in certain missions...).
At a bare minimum, if this has to be a "hire off the street" kind of thing, have them try to make sure that they are at least objectively hireable. I remember one spacefaring campaign where a player's new character couldn't explain any reason why our space freighter crew should hire him. In fact, he was pretty militant about it - we were looking for reasons to bring him on, since he was a PC... "So, what do you do?"
"Things... That need doing."
"Uh, do you have any specific skills we'd find useful?"
"I have many... Skills."
After about 15 minutes of that we took off and left his dumb ass on the planet. (We actually bounced the player from the group too, as being a muffinhead like that was a common character trait of his.)
I'd say all of Bruenor's major deeds and some of the minor ones, but only up until he left the throne for his final adventure with Drizzt.
So I'd say for a guideline the stuff that happened in the Crystal Shard.
The story about Bruenor and the black dragon when he first returns to Mithral Hall.
The war with the dark elves, and the war with the orcs and the frost giants, where he once again very nearly died.
The Treaty of Garumn's Gorge with the orcs.
These things I would say at the very least would be stories some dwarf obsessed with Bruenor would know. Although because your character is obsessed it would make sense for you to use obscure stories. Like the one about Lady Alustriel nursing him back to health after his scuffle with the black dragon.
I'd say the faking his own death wouldn't be common knowledge because only a few knew about it. Although this was common among dwarven kings, they would fake their own death for a chance to go on a final adventure pretty sure its mentioned in the series several times. Although I just remembered that the dwarves from Icewind Dale know about Bruenor's last adventure in Gauntlgrym so its very likely you could use this as well, as most wouldn't know it but someone who is a history buff could probably find out.
Best Answer
Most of 4e’s published (hardcopy) material was not Forgotten Realms and I don't think there are any specific 4e accessories for Amn. Fortunately for you, most of the 4e Living Forgotten Realms adventures are available for free online. While I don't know if there are any additional adventures based in Amn itself, as you noted there are a couple of Amnite colonies—Port Nyranzaru in Chult and Snowdown in the Moonshae Isles. Available Living Forgotten Realms adventures in the Moonshaes involving Amn and Amnian mercenaries are:
For more in-game historical background, there are the Forgotten Realms Campaign Setting and Player’s Guide for 3e, particularly the Shadow Thief of Amn Prestige Class in the latter.
In addition, AD&D 2e contains several Forgotten Realms accessories that directly deal with Amn, although these take place during the Time of Troubles in Faerûn before the Spellplague of 4e. Specifically,
You could also use material from Baldur’s Gate 2: Shadows of Amn, the computer game and the novel, which are centered in Amn and prominently feature Spellhold.
Dragon and Dungeon magazines have some Amn-related articles. For example, "Crimmor: City of Caravans" by Ed "I am Elminster!" Greenwood in Dragon #334 (pp. 22-34) is a loquacious, detailed treatment of the "Crossroads of Amn," including a keyed map, that was nominally for 3.5e but contains almost no mechanical information. (Courtesy of @HeyICanChan. Thanks!)
There are a couple of old 4e WotC forum threads on Amn. One, entitled "Heroic Tier Adventure Ideas for Amn," is self-explanatory and the other is "Meldread's Amn," one DM's detailed version of Amn about 100 years after the Spellplague, including maps.