Oh, it's Alucard alright. In his baby form.
But back to the topic at hand... the easy answer is Adrian was probably living the lifestyle a typical half human half vampire lives. It's been said that he was once evil like his father, but in the DXC version Death doesn't allude to anything like that, so is the original script still valid?
The way I understand Alucard... Adrian was raised under his father's tutelage, coming to build his powers (even though he was born with them). Lisa probably wasn't living with Dracula, because if she was, how the hell could that mob of people kidnap her without Dracula stopping them first?
Anyway Adrian was most likely still in contact with her. Somehowwww... the general populace were alerted to a 300 year old vampire that's been amassing all the powers of Hell without anyone knowing. Lisa, perhaps having atleast a little knowledge of magic due to having an extremely personal relationship with the Demon King, was killed and her husband was condemned.. that was when shit hit the fan and caused the rift between Adrian and his pops.
Immediately afterward Dracula wages war on humans, which causes Adrian to renounce his demonic heritage and switch his name to keep the promise to his mother. This is also when Hector "betrays" Dracula, Isaac flees, etc.
Then CVIII begins.