As someone that has done this before, here are some tips:
find a local printer that does sign/photo work and will work with you to match the color of a swatch of newsprint. Not to knock Kinkos, but they may not be the best place to go if you need to make a lot of special requests. If you print on bond paper (which is thin and low quality, like regular printer paper) you will have a print that will look just like newsprint and it will LAST. This is ideal if you are making a replica prop that you'd like to preserve. If you have the means to print on newsprint it will look great but once the paper meets ink, the appearance of the paper rapidly declines. In a year you will have a dingy, yellowed newspaper, in 10 years it will fall to pieces.
If you must print on newsprint, there is a resource called New York Fakery on the east coast that can print newsprint and also stocks a "newsprint tinted bond" that will outlast regular newsprint.
Large newspapers like New York Times, Wall St Journal, etc. print their proofs on tinted bond before they go to press. If you find a small local press that does newspaper printing it will be incredibly expensive and inefficient to run a single print.
If you attempt to run newsprint through your own printer, purchase "rough" newsprint, not "smooth" try to print on an inkjet as they can easily handle different weights of paper. In your overall design add a 10-20 % transparency layer of white on top of your design. If you do not do this, photos will appear too dark and bold black text will be muddled illegible on small fonts. If you look at a real newspaper, the blacks are more like dark grays anyway.
Xerox copy machines will demolish newsprint and immediately jam because the paper is so thin and fragile. Don't even try it, it won't work.