Learn English – The meaning and word origin of “Christian with four aces”

bane: a cause of great distress or annoyance

Therefore the bane of your existence is the chief annoyance or distress in your life, it is something that prevents you from enjoying life, turning it instead to misery.

From White Fang by Jack London:

But the bane of his life was Lip-lip. Larger, older, and stronger, Lip-lip had selected White Fang for his special object of persecution. White Fang fought willingly enough, but he was outclassed. His enemy was too big. Lip-lip became a nightmare to him. Whenever he ventured away from his mother, the bully was sure to appear, trailing at his heels, snarling at him, picking upon him, and watchful of an opportunity, when no man-animal was near, to spring upon him and force a fight. As Lip-lip invariably won, he enjoyed it hugely. It became his chief delight in life, as it became White Fang's chief torment.

Bane itself is an unusual word in English and outside of this common phrase it is rarely used in the modern language. According to phrases.org.uk it was present in the Old English Chronicles and meant 'murderer'. It can also be found in the common names of the plants henbane and wolfsbane, both of which contain deadly poisons.

It basically means don't pick a fight with the press or media.

It sometimes called "Greener's Law" and attributed to William Greener, but according to this page, it was Congressman Charles Brownson who coined it in 1964.