Firstly it's relevant to point out that a (UN-supervised) Kosovar Supreme Court ruling in 2001 said that Serbia had not committed genocide against ethnic Albanians in Kosovo - though it had committed war crimes and crimes against humanity.
This ruling was reportedly based on the 1948 Geneva Convention, where genocide is defined as intent "to destroy, in whole or in part, a national, ethnic, racial or religious group as such".
The ruling was made by two international judges and one Albanian judge, at a time when there were concerns internationally for the treatment of Serbs by the Albanian-dominated judiciary in Kosovo.
Some UN officials were very unhappy with the ruling that Serbia had not committed genocide. The ruling was also made while the excavation of mass graves was still ongoing. Nevertheless it is currently not considered to have been genocide.
Following Tito's death in 1980, Yugoslavia's federal authorities spent much of the Eighties trying to quell both Serbian and Albanian nationalism in Kosovo, which - to the chagrin of future Serb nationalists - had been made a province in 1974.
Kosovo's economy was faring poorly and, in 1981, when students protested in Pristina against their poor living standards, more serious rioting broke out as students allied with a local workers' strike. The riots assumed a Kosovar/Albanian nationalist angle and were eventually shut down by federal authorities declaring a state of emergency (Arhsien & Howells, The World Today, Vol.37, No.11 (1981), p.419).
A narrative emerged in Serbia that Serbs were being driven out of Kosovo, best captured in the publication of the controversial SANU Memorandum in 1986 by the Serbian Academy of Sciences and Arts (САНУ/SANU). While making broad prophecies of doom for Serbs throughout Yugoslavia, the memo specifically claimed that Albanians were committing genocide against Serbs in Kosovo (p.41, 56).
Slobodan Milosevic, then a leading member of the Serbian Leeague of Communists, joined federal government in condemning such "nationalism" from SANU. But by 1987 Milosevic was championing the cause of Serbs in Kosovo, and by 1988 had manoeuvred to become the President of Serbia.
There's obviously a lot more to it than this, but an overview of some key factors from years earlier.