I believe they claim descent from the Prophet Muhammed through his daughter Fatimah. The dynasty is Persian in origin but later lived in what is now Pakistan. Although many have been born in Europe. Both Queen Elizabeth and The Shah of Iran conferred titles of His Highness on the current Aga Khan due to his descent from the Qajar dynasty.
I know that the current Aga Khan's mother and both his ex wives were women of European origin some of whom converted to Islam. His mother was Joan Yarde-Buller who was the daughter of John Yarde-Buller, 3rd Baron Churston. She was previously married into the Guinness family and lastly became Lady Camrose, so they were connected to the British aristocracy/nobility there. His first wife was a British debutant who's first marriage was to a Marquess.
His grandmother also seems to have been at least partly of European origin, her name was Cleope Teresa Magliano. According to Wikipedia the 2nd Aga Khan had a third wife who was Persian nobility, so Prince Aly's Great Grandmother. The Aga Khan title skipped to Aly's son the current Aga Khan because his father the 3rd Aga Khan thought Aly's lifestyle was incompatible with being the spiritual leader of Ismailis and needing a young modern leader for his followers, which seems to have been a wise decision.
Both his oldest adult sons have married American women who converted with one divorce and remarriage. And his daughter Zahra married and divorced a British man.
Prince Aly Muhammed, the youngest son of Aga Khan IV has an older half sister Princess Teresa of Leiningen who's father is a pretender to the Russian Imperial throne and was declared as such by one of the Russian monarchist parties.
Edit: Or what Blog Real posted for ease.