I am a single individual on a student F1-visa from India. I study in the state of North Carolina.
For the year 2013, I had a job for some time in NY city, and have tax withheld for NY State and NYC local income tax.
For the same year, I also had a job in NC. I have tax withheld for the state of NC as well due to that.
I have already filed my federal paper returns. But, I am not sure how I file my state and/or city taxes to claim a refund.
Last year I just had income from NC so had to file the D400 for NC. But, this year I am not sure as to what I should do.
Can someone help me out? (My total gross income was around 20K)

you are a full year resident of NC, so you as a nonresident of NYS/NYC need to claim the taxes that you paid to nys/nyc on your NC return. only residents of NYC pay NYC income tax. Non-resident workers do not pay NYC income tax.

But I am classified as a non resident alien due to my F1 visa status. As per your reply, I don't need to file any returns for NY and NYC and the same amount withheld should be claimed in my NC tax filing?

As per my F1 Visa status, I am a non resident alien in the state of NC, does that affect the answer that you gave.
Just to be clear, what I understand from your post is that I only need to file returns for the state of NC and claim the amount withheld for NY and NYC in these returns?

#1; that is for your fed return; for youe state return, you, a F1visa holder, is treated as a full year residnet of NC state.

#2; as you said, as you worked in NYS/NYC(if more than 183 days) and earned incoem there as a non resdident of NYS, thenyou need to paytax on yur nYS source income to NYC/NYS however, as a full year resident of NC, you can claim your tax that you paid to NYS/NYC to your NC return