To proceed, you'll need new strings and a tuner.
New strings are absolutely necessary because finger jam and peanut butter encrusts old strings.
An old string is never equally dense or elastic along its length, and this will undermine any attempt to achieve perfect intonation. A high-quality tuner is necessary for the kind of precision we're after. Seasonal temperature and humidity changes can also affect guitar intonation. Put on new strings, taking care to remove any slack at the tuning pegs. If your guitar has a locking tuning system, you probably can skip this step.