Drangov was born in Skopje in Ottoman-ruled Macedonia in 1872, to the family of a rich timber merchant. In 1891, he entered the National Military Institute (today the Vasil Levski National Military University) in Sofia. In January, 1895, he became an officer and was promoted to the rank of second lieutenant, and by 1899, senior lieutenant.
In 1904, with the rank of lieutenant of the 2nd Cavalry Regiment, he was sent to Russia for further studies . Boris entered the Russian General Staff Academy in St. Petersburg. During his training in Russia he was promoted to captain. In 1907 he graduated with honors from the Academy of the General Staff in St. Petersburg and returned to Bulgaria. He later participated in the First and Second Balkan Wars.
In October 1915, Bulgaria entered World War One on the side of the Central Powers. Drangov was assigned to command the 5th Macedonian Regiment, 11th Infantry Division. By 1917, he was transferred to the Thessaloniki Front and appointed commander of the 9th Plovdiv Regiment, 2nd Infantry Division. He was wounded during artillery shelling on 26 May 1917 and died of his wounds the same evening.
He was buried in the courtyard of the Orthodox Church of Saint Dimitrija in Skopje. After the war, the Serbian authorities transferred his remains to the city cemetery. He was posthumously promoted to colonel by the Bulgarian Army.