Post History

Begin by noticing that educated is a relative term. Today we tend to think of it in terms of formal schooling. But many people with less formal schooling, may be educated in other things by other means, such as apprenticeship. Similarly, someone highly educated in physics may be completely ignorant of history, or vice versa. 

So the first question you should be asking yourself is, what does this person know, and what do they not know. Ask yourself what misconceptions are common among people not educated in a particular subject. These will be important clues that the character is not educated in the subject under discussion. 

Also, when people are struggling to understand something, they commonly try to cast it into terms they do understand. (A gifted teacher tries to do the same thing for their students.) So use how the character tries to express themselves in terms of their own expertise to indicate their level of understanding of the subject under consideration. 

Don't rely on bad grammar. Most people speak grammatically, even if they struggle to write. Their grammar may not always be standard, but that is a different thing. It is a matter of region or class variations from standard grammar, not lack of education. Use these, by all means, if you understand them well enough, but not as a sign of lack of education. 

Actually, it is the partially educated that are most likely to use bad grammar in speech, because they try to follow grammar rules they learned in school that are either incorrect , or that they don't understand, or are applying incorrectly. The uneducated don't make those mistakes because they were never taught these rules. They speak in the grammar of their people. 

There are some aspects of "formal" grammar that act as shibboleths, designed deliberately to set the nobility apart from the peasantry, but the use of these also has more to do with class than education. 

In summation: use local patios/dialect and their grammar if you truly understand it, and you think your readers will recognize it. But otherwise focus on what your characters know and don't know, and how what they know shapes how they inquire into the things they don't know. Education is a matter of knowledge and ignorance (when not corrupted by politics), so focus on what they do and don't know as markers of their education level.