Headcanon on Pre-Shadow Isles Karthus
Here are my headcanon theories on Karthus' life before he embraced death to become a Lich. This will involve his reworked lore and a bit of the Pentakill lore, omitting Karthus' death at the hands of Mordekaiser as the Deathsinger had canonically died at the end of his pilgrimage to the Shadow Isles.

I imagine that Karthus' past occupation was either doctor or taxidermist, since it was stated in his most recent lore that he tended to the sick and dying to closely witnessed their deaths and he had a hobby of preserving animal corpses. Perhaps his family's background was in medicine or taxidermy, which could explain why he had a such a strange fascination with the dead. Since Karthus was so obsessed with death and avoided the company of others, he was definitely introverted and even antisocial.
Hear my song I also think that Karthus had a knack for poetry and music in his life. He might enjoyed the death hymns that he might have heard the funerals he snuck into. In addition to talking to the tombstones of strangers, he probably sang to them, practicing the hymns he learned and created his own morbid songs. Since he has such a talent for singing, this made him a natural fit as the lead singer of the Pentakill band. However, since Karthus does still have a problem with people -particularly living crowds- that's why he drinks before he goes on Pentakill performances. Even in undeath, he despises the company of others, so alcohol serves the purpose of temporarily removing his disdain for the living.
Ah, the kiss of death When Karthus decided to fake his death, I had to wonder if that act was an misguided attempt to seek attention from others. After rereading the part when he had shunned the company of the living, it was apparent that Karthus had a disturbed desire to see how others would react if he passed away. Would anyone miss him? Shed a tear at that fact that this lonesome, macabre man had died? Perhaps tears were shed as he was mourned. In their eyes, Karthus may have been a eccentric, ghastly man, but he was still part of their community. His contributions to tending to the sick and dying were not forgotten. When his community found out that Karthus had staged his death, I could not help but envision their shock and outrage. Karthus was a trusted member of their community and he had betrayed their love and trust; he manipulated them as pawns in his twisted experiment.
It ended up with his community disowning him.
My pilgrimage begins After his staged death incident, Karthus most likely left his home, not because of the complete rejection of his community, but because he could finally set out now for a pilgrimage to the Shadow Isles. It was at this moment, I realized Karthus' brilliance. He had to sever his ties to the living in order to dedicate himself to death. I found that Karthus was similar to Zaheer, a character from Avatar: Legend of Korra. Similar to Karthus, Zaheer had to sever his own earthly ties (his beloved lover, P'Li) to be able to fly.
Have you ever felt a calling? Referring back to Karthus' situation, he wanted to give his life willingly to death, but he believed that he may have second thoughts and refuse to take his calling. Although he was most likely the black sheep in his community, he probably still had people who may have cared about him. So by staging his death and revealing that he was not dead to his community, he ensured that he had nothing to come back and absolutely no second thoughts once he had set out on his pilgrimage to seek out the Shadow Isles and consequently, Death.