Is "To get you to know me better" grammatically correct? It is supposed to be an informal letter.
to get you to know me better
is not, as pointed out, a complete sentence. It consists of one infinitive clause
- .. (for you) to know me better
as object complement to another infinitive clause (with causative get)
X) to get you [(for you) to know me better]
Their subjects, you in the lower clause and
X in the upper one, got deleted with the complementizers,
leaving only the bare subjectless infinitives. So, as far as it goes, it's a correct phrase.
But it's not correct as a sentence, because infinitives can't form sentences.
A sentence has to have a main verb that is inflected in the present or past tense.
And there are lots of sentences where to get you to know me better would not work.
Executive Summary: Use complete sentences when asking questions and when giving examples.
It sounds awkward and should be reworded to more accurately fit the context of the particular statement.
For example, to use a previous contributor's example, "I've sent you these pictures and some of my writing to get you to know me better," you don't want the person to know you better, you want them to know your pictures and writing better.
It would be better to say something like, "I've sent you these pictures and some of my writing so you can get a better idea of my skills."
Sometimes, rather than struggling over individual words or phrases, it is better to cross everything out and start over. Think about the message you are really trying to put across, and go fro there.