My rough chronology. Definitely open to modification, but I stand by it in general.
"Exle of Atlantis"/EoA
I think it's hard to argue that this isn't first. However, it has real inconsistencies with later tales that could be discussed at some point.
"Curse of the Golden Skull"/CotGA
REH calls Kull "of Atlantis" and a "barbarian chief". It's hard to see this being set after Kull took the throne.
"The Shadow Kingdom"/TSK
The fact that Kull hasn't met Ka-Nu/Kananu (or Brule) yet points to it being very early in his reign. In fact, the first couple pages are probably Kull's inaugural parade.
"The Mirrors of Tuzun Thune"/TMoTT
It all gets trickier from here. The fact that Brule is the only one around points to an early placement. So does Kull's extreme naivete.
"The Cat and the Skull"/TCatS
Once again, very naive. Plus, no Kelkor/Kelka.
"The Screaming Skull of Silence"/TSSoS
Kuthulos is back (and never again after). No Brule or Kelkor.
"By This Axe I Rule"/BTAIR
Kull says it's been "months and months" since he took the throne. Still no Kelkor around. Plus, there's still a Serpent temple around (though that doesn't necessarily mean anything).
"The Altar and the Scorpion"/TAatS
Kull cleaning house after the assassination attempt. Kelkor possibly shows up around this point.
"The Striking of the Gong"/TSotG
Another (suicide) assassination attempt. This could go here or after "Stagus".
The Stagus Fragment ("Sunrise")
To me, this marks a turning point. Kull is still making idiotic decisions, but things (appear) stable enough that Brule and Kelkor (and Tu) seem to feel Kull can just ride off to Valka-knows-where taking the best warriors of the kingdom with him.
The whole thing is pretty obviously a setup. Maybe Grondar, maybe the Triple Federation (Thule/Commoria/Kamelia) or maybe Thulsa Doom. The whole thing could've easily become a novel. I see the consequences of this finally making Kull see he's got to take the kingship thing seriously and quit haring off at the drop of a hat or sitting around musing. In the rest of the tales he seems a bit more Conanic.
The Black City/TBC
On my map, I place Kamula up near the Kamelian border, since I think it's reasonable that it was once part of Kamelia but annexed by Valusia at some point before Kull. Bad blood there against Valusia, at least in some quarters. This could be before or after the defeat of the TF (which includes Kamelia). Kull is obviously good buds with Brule at this point.
Swords of the Purple KIngdom/SotPK
Kull has fought the Triple Federation and Grondar. He's got things well in order. Kelkor. This story has to be placed later. Hence, here.
The Brule Fragment ("Wizard")
Things are settled, plus, we've never seen Ronaro before.
The King and the Oak
Very hard to tell what's going on here. Could go just about anywhere after BTAIR.
Kings of the Night/KotN
Thurian Age Gonar shows up. Never before seen. Brule gets the "Morni jewel". Kull possibly at his most Conanic.
Looks about right to me. You have to sort of follow how he wrote them since he seemed to add characters as he went along (C&S followed by Skull of Silence being one example). The King and the Oak is a perfectly fine way to end the Thurian portion of his life, to be capped off by KotN.