The history of the Hellsing family - the interesting bit, at least - begins with Professor Abraham Van Helsing, M.D., D.Ph., D.Litt, etc., etc., whose expertise in rare diseases is called upon in 1897 when Lucy Westenra begins suffering a strange affliction. She is, as Van Helsing realizes, being drained by Dracula.
It is Van Helsing's expertise in the occult that proves most useful. With his help, Lucy's friends fight back against the vampire and end up chasing him back to his castle. Here, Stoker's book says, they destroy him.
The book lies. Dracula has been serving the family ever since.
Though Van Helsing's wife went mad and his first son died young, he ended up with a son named Arthur, who served the English crown. During WWII Arthur (who had anglicized his last name to "Hellsing") was a hedonist who drank quite a bit, to the irritation of comrade Sir Islands. Nevertheless, he ran the Hellsing Organization.
His ace fighters in 1942 included Dracula, albeit in a far different form.
Around 1967, for whatever reason, Dracula was locked in a dungeon in the basement of the Hellsing house, and Arthur's younger brother Richard never knew anything about him.
Some time between the war and the present, Arthur married a woman of Indian descent, and in 1974 his only daughter - Integral Wingates Hellsing - was born. When she was thirteen, her father died, and declared her the organization's new leader.
Her uncle Richard objected. He expressed his objection by trying to kill her.
She remembered her father's emergency instructions, went down to the basement; and found Dracula.
And the rest, as they say, is history.