Just because I wanted to point out a few things.
First - most of the history that the Master had/has with the Doctor, Koschei shares with his. However, there are always differences. The fact remains that they never fell out the same way they did in canon, and this Koschei works 'with' his Doctor just as often as he works against him.
Due to their relationship being less strained and his not taking on the name 'the Master', Koschei still sometimes calls the Doctor Theta, or Theta Sigma, at times. Usually either when he's being all nostalgic and compassionate (rare indeed), or if he's angry.
Deviations from canon mainly start as of Utopia, though. This is because where in canon the Master immediately acted against the Doctor and his friends, Koschei did not. He still needed to regenerate into Simm!MasterKoschei, though, but I haven't figured out why or how that happened. He kept his TARDIS, which is a newer model than the Doctor's.
Events leading up to his intro at SWS are as follows:
He runs from the Time War, fobwatches himself into Professor Yana, and at some point meets Chantho.
The Doctor, Martha and Captain Jack find him in the year 100 Trillion, and he's de-fobwatched. Goes back to being Koschei.
Dies, regenerates in the Doctor's TARDIS.
Finds his own TARDIS, taunts the Doctor over the difference in Type, and leaves. He takes Chantho with him, fully expecting the Doctor to take care of the last remnants of humanity in some way.
He travels with Chantho for an as yet unspecified period of time before leaving her somewhere. During this time, it is assumed that he had some interaction with Lazarus labs, and inadvertantly caused the events of that episode due to his desire of the ageing circuits for his new laser screwdriver.
Then sets about wandering around, with no real aim. Just prior to SWS intro incites riot and revolution on Tessaria IV with the cause being a lack of freedom in music... or something like that.
While he's not about to do anything like the Master did to get Gallifrey back, Koschei does miss his home planet and the other Time Lords. He does not, however, miss the way that they treated him (pretty much the same way they did with both the Master and the Doctor - he's as dangerous as the Master, and even more rebellious than the Doctor), or how they dressed.
Companions - he's had them. He'll occassionally refer to it as keeping pets, but basically, it's the same as what the Doctor does - take someone on, usually not human but those do crop up from time to time, and show them the universe. The differences being that he does it to stoke his own pride and ego rather than to bring out the best in his companions, and that anyone who chooses to travel with him has to put up with his attitude towards them. Which basically includes a lot of rudeness and him talking down to them as if they aren't expected to know the simplest things.
Unless, of course, they're 'family' - he's taken one of the Doctor's daughters - Arianne Pond - on a trip in his TARDIS, and is leery of but tolerates a girl who's the daughter of one of his alternates. In this case, 'family' could mean anyone with Time Lord blood in them, especially since apart from him and the Doctor, there are no other Time Lords. Seeing as the drums are that much louder without that constant presence in his mind, he's going to take what he can, thanks.
And on that note - and going sort of more OOC here - if anyone wishes for their character to become crew/passenger? Just drop me a line somehow, and I'll see if we can sort something out.