At the start of the game, Kurdistan's territory is held by Turkey, France (French Syria), Iraq and Iran. This can be done as any of them (or any nation, really), but Turkey and France are the easiest, the former due to holding most of the cores from start and the latter due to being a major power. Whichever nation you pick, simply release Kurdistan and choose to play as them AFTER you've conquered all Kurdish cores.
As Turkey, this is easiest if you reform the Ottoman Empire ("Nobody's Business but the Turks" - see Filip36365's guide for help). After the Empire has been reformed, you'll be able to take the focuses "The Damascus Diktat" to demand Syria from the French, "Expand the Saadabad Pact" to form an alliance with Iraq, Iran and Afghanistan, and "The Pan-National Association of Ulemas" to puppet your allies in the Saadabad Pact. If France accepts your demands, this secures France's Kurdish holdings peacefully, whilst if Iraq and Iran agree to join Saadabad and become your puppets, you can simply work them down until they're ready to be annexed. If any of these fail, you'll have to conquer the territory the old-fashioned way. Once you've secured all territory, do as I said before and release Kurdistan, choosing to play as them.
As France, this is easiest if you flip ideologies away from democracy, as both communism and fascism have unrestricted war goals, and non-aligned only requires 50% world tension to justify a war, as opposed to democracy's 100% to actually declare war. After flipping ideologies, you should probably seek to defend yourself from Germany first of all, especially if you're playing with historical focuses on, as you'll most likely be attacked by 1940 the latest (no guarantees, as France isn't targeted by Germany's focus tree despite holding territory Germany historically wanted to reclaim). Once your continental borders are secured, you'll want to declare war on Turkey, Iraq and Iran. There's a chance Britain will guarantee them or they'll enter an alliance, especially if you are playing non-aligned and the world tension is already at 50% or higher. Conquer them quickly and annex the Kurdish cores, doing whatever you want with the rest of the countries. Then, as before, release Kurdistan and choose to play as them.
As anyone else, you'll want to follow the same guide as France, flipping to either fascism or communism preferably (if you weren't originally) and declaring war on the nations who hold Kurdish cores.