In the game Faustin has Niko kill Lenny, Kenny Petrovics son. He believes he is a rat but no real evidence proves this about Lenny. Now later Dimitri tells Niko that Kenny wants Faustin dead for this and as long Niko is the one to kill him, Niko will be spared. After Faustin is killed Dimitri betrays Niko.
So if Dimitri is telling the truth Petrovic understood Niko was just a hired gun, so he had to pay for the killing of Lenny by killing Faustin who ordered the death of Petrovics son. Then with Faustin dead Dimitri took charge of the Russians and was able to contact Bulgarin as a new associate. Dimitri talked about Niko helping with Faustin, Bulgarin recognizes Niko being his enemy, Dimitri prefers a powerful friend like Bulgarin over Niko so he betrays Niko for Bulgarin.
Then again Petrovics gang had presumably been very hostile with Faustin and his Russian Mafia. I would see that with Lenny being killed that Kenny would not care if Niko was a hired gun and so Kenny would want Niko and Faustin dead for sure, Dimitri too for being close to Faustin and being Faustins second in command. Petrovic would might as well just wipe out the remaining Faustin russians at this point. Now I don't know how Dimitri would stop Petrovic getting revenge, but he might of used the Lenny killing to create the story he told Niko so he would kill Faustin for him, and Dimitri would take over. As Dimitri saw Faustin becoming mad and Dimitri was always out for power and betrayed anyone to get it, so he figured Faustin was now insane and made up a story to get the deed done through Niko.