I bought an out-of-business competitor's domain, which still ranks in Google. What is the best way to use the domain?

Get it appraised then either hand on to it, start a new site to direct it to and build up the asset or just list it for sale.

SEO isn't my area of expertise, really. But in other respects, domain names are. So I'll throw out a few ideas for you to think about.

It's possible the domain name itself has market value apart from the back links, customer memories, and residual traffic. In that case, you might list the domain for sale. However, it's impossible to judge without knowing what the domain is. Most likely there is no intrinsic value. Just the SEO or lead-generation benefits you might experience as a business with an adjacent service offering.

Rather than redirecting the domain, you might consider using it separately. There are a few ways to do this.

Maybe there's value in developing their old domain as a secondary website for your business. If their domain is descriptive of what you do, this might make sense. However, if their domain simply represents their defunct brand, then developing it would actually conflict with your own brand identity.

Another option is to put up an efficient single-page lander on their domain. It might explain that the old company has gone out of business and funnel interested customers toward your website, either through links or a contact form. Whether this is worthwhile or not depends on the quantity / quality of traffic their old domain is getting. How many leads would you expect to receive and what are they worth?

Before you do anything like this, however, you'd want to think about legal repercussions. Maybe there aren't any. However, it's possible the company went under, the domain expired, but the trademark remains active. That could pose a risk if you use their old domain without permission. Assuming they object and have the funds to sue. If there's a lingering trademark, you might want to have a conversation (in writing via email) with the old owners about their future intentions regarding the brand name.

Just some options. If you'd like to discuss further, I'm happy to chat.

It depends, if all the organic traffic is going to the home page and your domain target the same topic, you woud probably transfer the ranking. If there are lots of pages ranking on the domain you bought, you would need to offer the same content on your site and redirect url by url!