Your only chance to join the Aldmeri Dominion is during the first quest.
The Imperials and Stormcloaks will no longer be hostile once the war starts. If Solitude/Windhelm is controlled by the Stormcloaks/Imperials, or once Tullius/Ulfric has died. You will receive a letter soon after the civil war has ended.
If you have any problems with it from installing during a play-through, let me know. It appears to work fine with a save-game installation, a new game is recommended though. The fate of Tamriel will soon be decided, which side will you be on? With the Empire in ruins and the people of Skyrim already beaten and bloody from war, can anyone stand up to the might of the Dominion? The Aldmeri Dominion, never ones to miss an opportunity, declare war once again. The Empire is as weak as it has ever been. The Second Treaty of Stros M'Kai was signed and the war was finally over. The Redguards continued to wage war against the Dominion for another 5 years before they finally succeeded in pushing them out of Hammerfell. This would lead to lasting bitterness between Hammerfell and the Empire. The Redguards of Hammerfell objected to this, and so the Emperor renounced Hammerfell as an Imperial province.
Per the terms of the treaty, the Empire was to secede a large portion of Hammerfell to the Dominion. This lead to The White-Gold Concordat being signed, its terms heavily favoring the Dominion.