The Second Balkan War was an unfortunate, but natural continuation of the First Balkan War mostly due to the newly liberated (current border) regions with mixed population and due to the strong influence of the European Great Powers, i.e. United Kingdom, Germany, Russia, Austria-Hungary which were struggling to establish and realize their strategic interests in the region (Balkans).
In the mixed regions of Eastern and Western Thrace and all Macedonia with Thessaloniki, Edirne there were conflicts that were part of a wider rebel war from the early 1900's, called the Macedonian Struggle, between Bulgaria, Greece and less by Serbia, settled by a Serbian - Bulgarian treaty signed in 1912.
Additionally Romania was not satisfied with the signed treaties regarding the land separations between countries of the Balkan League and saw opportunity to acquire Northern Dobruja.
Finally, Turkey (by the time still Ottoman Empire) acted opportunistically as all the Bulgarian Army was concentrated on the South-Western/ Serbo-Greek border and the result of its invasion, in violation of the Treaty of London 1913, was the recovery of Edirne (Adrianople).
These were the unofficial relations between the Balkan countries at the end of the First Balkan War. Additionally there was a secret defensive alliance between Serbia and Greece, signed on 19 May/1 June 1913 that was aimed to preserve the occupied Macedonia. Montenegro just joined after the start of the war.
To conclude, the Second Balkan War was a conflict expressing the discontentment of the counties of the Balkan Leagues regarding their new territories. Ferdinand I of Bulgaria started the war as he felt that he deserves more land1, firstly due to the San Stefano Peace Treaty 3 March 1878 according to which the mixed regions were within the Bulgarian borders, moreover, within the borders of the Bulgarian Exarchate. Secondly, due to the fact that the Bulgarian army lead the main military conflict against the Ottoman Empire in the First Balkan War, suffering the most loses, army which was equal in numbers with the sum of the armies of the rest of the countries of the Balkan League (Greece, Serbia and Montenegro). There were multiple violations and war crimes committed by all sides that doomed any future constructive relations, described in the report of the International Commission with title "Inquiry in the Causes and Conduct of the Balkan Wars.".
1. Ferdinand I of Bulgaria envisioned Greater Bulgaria which was similar to the Greek Megali Idea, both of which were a chimera, but most importantly mutually exclusive and lead to national catastrophes with ethnic cleansing, large migration waves and territory reductions.