My dad has a company that has been passed on from generation to generation that I was supposed to inherit, but that's where the problem lies.
A few years ago I went down a dark path and completely lost myself in theft and illegal substances and went to jail for a while.
After I came out I started my life over from scratch, got a good girlfriend and found some new friends.
I recently got back into contact with my dad and it turns out we share a lot of hobbies such as politics and economics.
Now I really want to work in my dad's business and eventually claim the company for myself and continue his work, but my dad is a little weary knowing what I used to be like.
How can I show my dad that I'm serious and am eager to learn from him?
Your only chance at dissolving any suspicion between you and your dad is to spill your guts on the subject in an earnest and composed fashion. Sit down with him and tell him you want to work for him, let him know you are aware of all the reasons he has not to let you into his company. Tell him you used to be a really shitty person and it took you a while to wake up from it and get your shit together. Then sell yourself on your good qualities. Are you hard working? are you financially stable? are you sociable (particularly able to be socialable with strangers, talk to anyone and be on amicable terms) Tell him you want to work hard for him and if its still an option take over the business some day and continue his work. You're his son and he's father, you're gonna have to reestablish that connection with a clear cut discussion of what you want and what you're willing to do to be trusted again. And at the same time refrain from using emotional leverage to get what you want because he will be able to pick up on it and it will be an indicator that you can't be trusted.
How big is the company?
50 employees? 100? 200?
If it's just a shop it should be fine but if it's something large you represent a serious image problem
You probably need to get married have two kids, become a devout christian and become an outstanding member of society