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IIRC the king's wife starts the quest to go to Alik'r. It should be right after you pick who becomes the ruler of Rivenspire. After you have that conversation with her, a quest marker will appear in the north-eastern part of the map at the docks. You can use that to get to Alik'r at no transportation charge.
Can someone tell me please where the quest giver is? Or do I have to finish Doomcrag first? I've tried two times and can't beat the Gargoyle.
SPOILER ALERT
You don't have to defeat the Gargoyle. Notice the statues in the room (4 of them). Stand behind one and wait for the gargoyle to do his attack. Move away and the Gargoyle will destroy the statue. Do that for all 4 and the gargoyle will die instantly.
I found this out accidently because i like to hide behind stuff.
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"I am a nightblade. Blending the disciplines of the stealthy agent and subtle wizard, I move unseen and undetected, foil locks and traps, and teleport to safety when threatened, or strike like a viper from ambush. The College of Illusion hides me and fuddles or pacifies my opponents. The College of Mysticism detects my object, reflects and dispels enemy spells, and makes good my escape. The key to a nightblade's success is avoidance, by spell or by stealth; with these skills, all things are possible."
One small addition. You can travel to Bangkorai (higher level than Alik'r so be careful at level 30) from the bridge, to the south-east of Stormhaven.
I was able to complete a couple of early quests here before spending much time in the Alik'r. It places a few things out of sync, but not so as you'd notice much. The upside is that I took a couple of early Skyshards and rose up by two levels, which made Alik'r slightly easier.
NPCs here are around level 37-38. Nice challenge if you're at 30