What factors related to the Eastern Crisis contributed to the outbreak of World War 1?

I don't exactly know what you mean "Eastern Crisis". There were lots of conflicts in Eastern Europe since the war in 18th century, just to mention Crimean War, war of 1878, two Balkan wars and maybe Italo-Turkish war of 1911-12. Also we should not forget Austro-Hungarian occupation of Bosnia in 1908 and something that is called "Pig War".

My opinion is that all these events had (alone) marginal effect on the outbreak of the Great War. As Francis Ferdinand was assassinated in Sarajevo, all Powers believed that the potential war with Serbia would be a local one, like the others I mentioned. The fact that it became a global war was in no way related to Austro-Serbian conflict as well as poor Archduke and his wife.

The only question between Ottoman Empire and Russia laid on the Straits and naval access of Russia to the Mediterranean. This was in fact not welcome by anybody, of course Germany and her southern ally, but by United Kingdom mostly. Russia in the east was a danger to British India and Egypt. That's why Cyprus was occupied. We can mention that during the Russo-Japanese war in 1904-05 Russian fleet was not allowed to travel through the Suez Canal. Persian agreement between UK and Russia was a compromise, the same in Afghanistan, it was strong because both parties gained what they want and secured it. However I think that UK would never agree to total destruction of the Ottoman Empire.

The Balkan question was seeking by Russia a way to Serbia and thus to the Mediterranean. Did it had impact on the World War 1? I don't think that much. The main winner of two Balkan wars (except Serbia) was Romania. This country could look for Russian assistance against Hungary and her occupation of Transylvania (and she finally joined the war on Allied side), but in fact she was independent. Romanian king was German. Serbia gained power, but it was still local power as separated from Russia.

So answering the question, the crisis(es) had their impact on the origins of the war, but in my opinion each single one was marginal, however the effect was cumulative. Balkan wars were just shuffling the cards; if you read Serbian response to Austro-Hungarian ultimatum in fact Serbia agrees to all Austrian demands, that is amazing, how humble she was. If Austria-Hungary had accepted it, Serbia would be almost politically destroyed. All the gains Serbia got during Balkan Wars would go away. All Powers (including Germany) agreed that this response should have been satisfactory; maybe Austrians feared other, 3rd Balkan war, and wanted to destroy her before? As Machiavelli said, if you fight with somebody, you should finish him, because if not dead he will seek opportunity for revenge.

There are two ways of looking at the Eastern Crisis as trigger for World War 1.

1. The brooding war in the Balkans caused unrest and hostilities expanded out from the region until a critical point was reached. Hostilities reached a tipping point when political figures begin getting assassinated (i.e. Franz Ferdinand heir to Austria-Hungary throne). This caused a cascade of international backlash.

2. European nations took sides in the Eastern Crisis, forging alliances based on self-interest. As sides were formed, hostility between the factions grew as war in the Balkans ensued.

The Eastern Crisis was certainly one of the factors that led to World War I, however it was not the only one.

• On the one hand, clearly the decay of the Ottoman Empire caused both Austria-Hungary and Russia to seek control over the Balkans. These two countries started WWI, mainly over their interests on the Balkans (with the assassination of Franz Ferdinand as casus belli).
• Yet the other countries joined the war willingly despite having no such interests. This cannot merely be explained by their alliances, they rather had their own interests. This is especially true for Germany that organized itself as a country merely a few decades before that and wanted to be perceived as a great power in Europe (seeking a "place in the sun"). A war was a chance for Germany to destroy the current balance of power and to establish a new one where Germany would dominate. If you look at the German pre-war foreign politics, there was a long series of conflicts with the British Empire and France and I would argue that Germany would end up at war with them one way or another.
• France was waiting for a chance to take revenge on Germany ever since the Germans beat them in 1871. In particular, they wanted to take back Alsace and Lorraine (something that they did after WWI) which are territories both Germany and France historically claimed their own.
• The British Empire defended its status of the strongest empire in this war: if there was one thing that the British absolutely couldn't use it was a strong German Empire dominating Europe.

So while the war started over a conflict on the Balkans, the reasons for the most countries to join it were entirely different ones. The main destabilizing factor in Europe was the newly unified German Empire which strived to change the balance - something that the established powers obviously opposed.