Note: This history was originally written in response to a letter from a soldier currently serving with the 69th NYARNG. It was written "off the cuff;" please comment if you have additional information to share on Co. B.
The origins of Co. B are the same as the rest of the 69th. The regiment was reorganized along with all other New York militia units in the 1850's so it would have been at full strength in 1861. Most maneuvers during the Civil War were at the Divisional level so Co. B would have moved with the rest of the regiment. Even if a platoon or two were thrown out as skirmishers during a particular engagement, it still would have been as part of the larger battle line (skirmishers generally served as a screen for the main line).
My group focuses on 1862, and I can tell you that the captain of Co. B, Thomas Leddy, briefly became the commanding officer of the entire regiment at the Battle of Fredericksburg. (The colonel and major were both wounded, and Capt. Leddy was the senior captain in the regiment.) He was quickly wounded as well, for the second time, and after convalescence he transferred to the Veteran Reserve Corps. He had previously served as a first sergeant in Co. K, which had its own colorful history.
Co. K was originally a separate unit, formed by Thomas Meagher, an Irish revolutionary who escaped a penal colony in Australia. He formed his own unit, the Irish Zouaves, because he liked to be fancy. (No, really. They had special uniforms.) His company was combined with the 69th regiment and became Co. K. Thomas Leddy rose through the ranks to first sergeant of Co. K before becoming the captain of Co. B, which means he probably had a personal relationship with Thomas Meagher, who was one of the greatest heroes of the Irish Brigade. (In fact, Meagher was colonel of the brigade at Fredericksburg when Captain Leddy was wounded, and resigned his commission afterwards because the federal government refused to reinforce the badly depleted Irish units. Meagher was convinced that it was racism toward the Irish units.)
I can tell you that they weren't called "Bravo Company," because the "radio alphabet" wasn't coined until WWI. They would have been referred to as Company B. I can also tell you that Co. B was recruited from New York City, presumably from the Five Points neighborhood in Lower Manhattan, right where the draft riots occurred.
During WWI I believe Co. B was commanded by a Jew, which was unusual for the Irish heritage unit, but maybe not so unusual for New York. I forget his name, but he was well loved by the men of the unit and there's even a song about him somewhere. (Obviously the unit has become much more multi-ethnic over time.)
That's about all I can tell you about Co. B in particular. As I said, the history of the 69th during the CW is the history of Co. B, all the way to the Appomattox Courthouse.