How can I extend my 60-day visa in Thailand?

If you need to stay longer than the extension allows then you can simply do a border bounce and re-enter Thailand on a 45 day exempt (30 after March 31st). That can also be extended for 30 days.

Assuming you arrived with a tourist visa you bought before you came to Thailand and were stamped in for 60 days when you arrived. If that's the case, then you can apply for a 30 day tourism extension at your local immigration office. If you're in Bangkok, this would at Chaeng Watthana. Here is a list of common requirements for this extension. You can only apply for this extension once based on your tourist visa entry, and after this extra 30 days you'd need to leave Thailand.

*Please note if you read this and you're also in Bangkok, Chaeng Watthana is only where you go if you had a visa when you entered. If you entered visa exempt without a visa, then you apply for an extension at the IT Square LakSi office.