How can I apply for a residency permit in Greece after entry and will a previous conviction impact my chances?

There are some kinds of residence permits provided in Greek immigration law, for which the applicant can apply directly in Greece under the condition that he/she had legally entered the country (e.g. as per the 90 days of your American passport). Such residence permits are the 3 alternative Golden Visa options. For the applications submitted in Greece, no criminal record certificates are required. If your application is related to other kinds of PRs, such as FIP, then it has to be submitted in the USA, through Greek Consulate and National Visa-D route. For National Visas D issuance a criminal record certificate has to be provided as well.

You can apply if you prove an income of more than 2k per month from rents, interests, pensions or other stable sources except for job/business.

You need to apply as a financially independent person.