Any hints on how best to start Byzantium in 1399?
It is easier, but not crucial, to start a few years later when Ottos are getting thrashed by Timurids. But what's key is to declare war right at the start on all your greek neighbors at once. Additionally, you need to declare war on Venice. All these DOWs need to be on day 1. Do this so that Venice can't come to war on it's vassals behalf because it's already at war with you. That way you can wage a war of reconquest against each vassal individually. Venice takes a little time to build up it's military, so your initial navy can defeat them and allow you to use single regiments in siege against greek minors after you defeated them.
What's tricky is that naples is probably going to declare war on the greek minor to the northwest (I forget the name). Taking on naples early is probably not worth your time. I recommend just letting them take it and using the choatic first years in more productive areas, chewing up minors.
With your starting 2-3 territories plus 3-4 greek minors, you should be able to become a medium strength power within a couple years. And you need to do this in a couple years. Because before you even sign the annexation treaties with the greek minors, you need to declare war on the turkish minors and start gobbling them up before the Ottomans get their acts together or they consolidate too much. Wipe out their armies, siege and keep starting wars as fast as you can.
Eventually, after 5 years or so, you should have conquered about 7 Greek and Turkish territories, all at a cost of 0 infamy. You should have never engaged in a substantial fight up until now. Then comes the big one, the Ottomans. They are still going to be bigger then you, so some finesse is important. Your first inclination is probably to use rebels, but I find that those should wait until later. If the Ottos are at war with someone else, obviously you just declare war in the confusion and wipe out their armies, but that's not likely. I find two strategies are best taking them on 1 on 1:
1) Go for the head. If they attack you in Constantinople across the strait, they are gonna have a bad terrain penalty and you can rout them easily if your numbers are nearly as big as them. They will retreat across the strait. Send your fleet out after the retreat starts to interrupt them (then send it immediately back to port to avoid it getting crushed) and they will stop, letting you attack them again. Once their main army is routed, quickly siege all their balkan possessions and use whatever you can to keep their new army in turkey. Wait a few years for the war exhaustion to skyrocket.
2) Go for the body. Often their army is in the balkans. In that case, they can't get at any part of you except for Constantinople, with it's level 2 fortress. Send your army to turkey, siege all their territories at once and leave a force just across the strait to dissuade them from assaulting Constantinople. At the same time, gobble up any turkish minors you missed so you control then entire penisula. Now, either cross the straits and move your army away from Constantinople to lure them into an assault to take down their morale and crush them or let rebels finish them off. This is my preferred strategy.
Once that's all done, you can usually take enough territories from them to split them in two and let rebels bring a few more territories into the fold. Ottos should no longer be a significant power and you can finish them off five years later. At this point, you look to spread your wings and start rebuilding the empire of Justinian and cast down the "pope" in rome!
Important in all of this is mint as much as you need to but do not collect war taxes. Your goal is to win a straight decade of uninterupted war. You do not need to do anything to increase your war exhaustion. Put all your research into government and get military drill for your first idea. All in all, pretty challenging, but within a decade, the Roman Empire can once again be the greatest nation on earth.