Drona, popularly known as Dronacharya, was a Brahmin, and had a childhood friend Drupada, who became a king. Once Drona went to Drupada and reminded Drupada of Drapuda’s childhood vow, of sharing some wealth with Drona. Hearing that, Drupada laughed out and said, that was a vow taken in child hood, when they were immature kids and now Drupada cannot do so, as he is a king. As a king, he can't share his wealth with a lower caste person. These words of an old friend hurt Drona and he angrily departed from there with revenge in his heart.
After that, Drona met Pashurama, a man well verse in all Vidyas. Pashurama was against the way, the kings were running their kingdoms. Drona asked Pashurama to teach him warfare techniques and Pashurama did so, only if, Drona took a vow of not ever sharing the Vidyas with any king.
Under the ages of Pashurama, Drona, learnt all the arts and became the greatest warrior. But soon, he broke Pashurama’s promise and became a teacher to Kauravs and Pandavs. He asked one part of Drapuda's kingdom as his Guru dakshina.
Drona made Ashwathama the king of the kingdom he got. By dividing the kingdom, Drona started a set of events: Drapuda prayed for a son, who would not shy from killing Brahmins and he asked for a daughter, who would divide the Pandava family property.
Now the question arises, whether Lord Krishna was right in killing Dronacharya, by beheading an Elephant, named Ashwathama and asking Arjun to rain arrows on Drona as soon as Drona heard about the news of Ashwathama’s death.
Drona did three grave mistakes:
1. He broke the holy Vedic caste system and became a warrior from a Brahmin
2. He broke the promise he made to Pashurama
3. He made his son a king, once again broke the caste system and showed favouritism.
Apart from that, there were other points too, the story of Ekalavya - a warrior. Ekalavya, was a tribal boy and Drona didn't teach him Dhanush Vidya, on the grounds of Eklavya's tribal background. Moreover, when Eklavya learnt Dhanush Vidya all by himself, just by erecting a life size statue of Drona and imbibing inspiration from it, Drona asked Eklayva to give Guru dakshina - Eklayva's Thumb.
Yet another crime is committed - demanding guru dakshina for teachings that were never taught by him.
These could be the main reasons, why Lord Krishna didn't shy away from killing an unarmed man, Brahmin and teacher.