If the Dragonborn speaks to Captain Aldis while wearing Stormcloak Armor, he will respond with: "What are you doing, dressed like a damn Stormcloak traitor?" the Dragonborn can either respond with, "Sorry, I didn't mean anything by it." and he will warn the Dragonborn not to wear it in the future, or "I am a proud supporter of Ulfric Stormcloak. What of it?" He will then say, "Is that right? Well this is how the Empire deals with rebel scum." The Dragonborn immediately receives 40 bounty and the captain and all the guards will attack. Killing the captain will not result in extra bounty since it is technically self-defence.
If the Dragonborn interrupts the execution in Solitude, all the guards will attack and Captain Aldis can be killed for no bounty.
Following the completion of the Stormcloak questline, Captain Aldis disappears, likely meaning he is supposed to have been killed in the battle for Solitude.
During the quest Bound Until Death, Captain Aldis; Pantea Atea; and the entire Snow-Shod family will likely attack you and be killed. Sadly, the mage section for the Bards' College will fail.
If you have completed the questline for the Stormcloaks, Captain Aldis will not be where the quest arrow is pointing, since he is an Imperial Soldier, and all the Imperials are gone.
This is confirmed, Only solution is to attempt the quest before liberating Solitude.
One solution is to open the console with '~'(tilde) key and type: player.placeatme 00041FB8
Be careful where you use the code. The Stormcloak Guards will attack him immediately. You can summon him in Angeline's Aromatics and complete the quest line there. He will die when he goes outside and the Stormcloaks spot him.
Quest is still available even after the liberation of Solitude. Miscellaneous/ Ask Aldis for information about Angelines daughter. Quest Starter> (Angeline's potion and ingredient shop in Solitude)
Also a notable bug: he can be found dead outside the castle door (this can be due to a dragon attack on Solitude).