I did a run like this not too long ago. Here's some basic steps to get you going in the right direction.
To start, you are going to want to vassalize Byzantium. Whether that takes 1 or 2 wars, it doesn't matter as long as you vassalize them. After that, use the diplomacy screen to see where the remaining byzantium cores are. Venice has a few, so your next major target is Venice. Ideally, they call in the knights. Peace the knights out first and demand annexation and then sell the province to the Byzantines. Have Venice return its 3 (I believe) cores to the Byzantines. Next, trebizond return cores to the Byzantines. Then, look for your own cores and take back all of those. Annex the Byzantines. Take the mission to conquer the Mamluks. Take a few provinces in the Sinai peninsula and at least 1 province that is a syrian core. Release syria as a vassal. Keep eating the Mamluks until you have fully annexed them.
At this point, you can go in several directions. Since you want to unify Islam and expand East... Diplovassalize your way across the north of Africa and into the Arabian Peninsula. In the mean time, DOW Qara and take 1 province from Persia core and 1 from Iraq core. Persia is going to be the key to taking down the Timurids. If Persia gets released before you can release it, it is going to be significantly more difficult to take down the Timurids and Persia itself. The Timurids are going to be your first real challenge. Just bait them into attacking you in the mountains and you should be fine. Take 2 Provinces (1 for Khorsan and 1 for Baluchistan) and then proceed to feed vassals. In the mean time, you should have been able to vassalize and annex Tripoli and possibly Tunisia.
Ideas: Start with expansion. The extra diplo relations is good, but the permanent CB in Asia is where its really at. You don't even have to border countries you want to attack. Next, either Offensive or Religious. Eventually, military points will be burning a hole in your pocket so it may be a good idea to take Military first and Religious later. Religious is necessary for expanding into Europe reliably later as well as reducing AE when complete.
One final note: for getting Sultan of Rum achievement, as soon as you can diplo-vassalize Crimea, do it and integrate. Attack Muscovy while they are still weak or you may find yourself in a difficult situation.
Hope this helps!
Here's an SS of what I managed to achieve using this method. I abandoned the game when 1.3 came out due to issues with vassal integration overseas. I never picked it up again when it got patched.
This was before Ottomans got overseas Expansion CB. I know I could do this much faster now.