"I am learning Turkish by myself or independently " How could I say this in Turkish. Thank you.

My try:

Türkçeyi tek başıma öğreniyorum.

My try:

Türkçeyi kendi kendime öğreniyorum veya bağımsız olarak.

Thank you Reyhan and Toggle

What does the "yi" in Türkçeyi mean? Thank you all.

It is the suffix that shows that ´Türkçe´ is the object of the sentence.  Suffix is ´i´ but you needethe buffer ´y´ here because the word ends in a vowel.  

An example without the buffer ´y´ is ´Seni seviyorum´ - I (subject) love (verb) you (object).

Brief explanation but I hope it helps