To what extent was Austrian and Russian rivalry in the Balkans to blame for the outbreak of the First World War?
Austro Russian rivalry in the Balkans only partly explains the reasons for the European conflict of 1914. It is true that they had been in conflict here before in the Balkan Crisis of 1908 and Russia was keen to avenge the humiliation. However neither side sought to escalate this to a wider crisis and wanted any conflict to be localised to the Balkans. Their local conflict was hijacked by the German Empire who sought to turn it into a much greater conflict that would involve a war with Russia and its ally France. Therefore the most convincing argument about the causes of WWI lie in an analysis of Germany's actions and less so with Russia and Austria.

Austria Hungary saw itself as the natural successor to the Ottoman Empire in the Balkans. To Franz Josef, the Balkans was an easy option to exploit. However amongst the many peoples of this region there was an aggressive and ambitious group who embraced Slavic nationalism and wished to assert Serbian independence from any imperial power but notably Austria Hungary by the turn of the 20th century. These nationalists had a much more in common with Russia who was equally interested in extending its interest in the Balkan region because of recent humiliations e.g. Russo Japanese war and for warm water ports in the Mediterranean.
You should add examples of:
Conflicts between Austria and Russia in the Balkans in 1908
Balkan Wars 1912, 1913
Assassination of archduke Franz Ferdinand

However these factors explain why a local conflict happened in 1914. They are less important in why it became a European wide conflict. Europe had been a cauldron of tension and rivalry for decades and the prospect of a European war had been the subject of much debate.
You should explain how the following created such tensions and rivalry
Anglo German Naval Race
Weltpolitik, German militarism, the personal role of the Kaiser, military plans, the Fischer thesis.
the role of the alliance system
domestic problems
Imperial Rivalry in Africa - Moroccan Crisis

These factors help to explain why so many major powers were hostile in 1914 but it again it does not explain why it boiled over into full scale European War in 1914. The events of July Crisis 1914 were much more significant in turning this regional crisis into a European wide conflict. Germany saw an opportunity to fulfil its long term ambitions of dominating Europe. It realised that a war sooner rather than later was in Germany's best interests and encouraged Austria to make a bold stand against Serbia over the assassination of the Archduke. Germany knew it would draw Russia and France into war. The Schlieffen Plan would then be put into action. The Kaiser and German military were highly confident GB would not get involved and that victory would be swift.
Austrian Ulitmatum, Blank Cheque
Moltke Memo and Bethmann Hollwegg
German Mobilisation
Kaisers gamble over GB - ‘contemptible little army' ‘scrap of paper'

Austria was manipulated by Germany into its aggressive action in the Balkans. Russia on the other hand tried very hard to stop the war escalating as the telegram exchanges between the Kaiser and the Tsar show, although it is worth pointing out that the Russian generals were encouraging the Tsar to fight. By July 30th 1914 war in Europe between most of the major powers was inevitable. Germany wanted it and engineered it. As hard as GB, and the Foreign Secretary Edward Grey, had worked to avoid war, after this date GB had no option but to go to war or else risk the permanent balance of power in Europe shifting in Germany's favour. Every major port in Europe would be under German control. This was unacceptable to this island nation as GB's Empire and economic interests would be at risk.

Although it may appear that the long term causes of tension simply bubbled over into war in 1914 as Lloyd George claimed in the 1930's, it was not inevitable until Germany began to manipulate the regional crisis in the Balkans between Austria and Russia. The German military knew that if war was delayed any longer then the Russian and French forces