I contacted the owner about a property that was advertised as vacant. However, when I went to view the property there was an entire family that had just moved in two days prior to me coming to see the property. I was told the previous owner had signed agreement with the tenant for only $650/month. I was very upset and contemplated backing out of the deal, but the owner (who is a private lender) was willing to do an owner finance deal with me. I saw this as a benefits due to this  gives me the opportunity to earn passive income without the debt showing on my credit. Terms of the loan with the lender/seller:

Sale price: 45,000

15% down at 9.85% amortized over 30 years.

My question is: Can I update the lease agreement the tenant has with the previous owner?

You won't be able to change the lease since it is a legal document.  You would be violating their rights by changing the lease after you own it. 

If the owner is smart, he will file a lien against the property since it is financed.  I think that will trigger the debt on your credit report.

You might be able to add an addendum, but you sure aren't increasing the rent.

And this loan will in fact show on your credit.