It's simple, just go to Journals & Newspapers, type in the title of the newspaper and press return! If we have access to it via a database, the database name and dates of coverage will be listed.
Other libraries may have copies of other newspapers. To search other libraries use these resources:
Contains over 48 million records, covering all materials cataloged by OCLC member libraries around the world.
If you want to find names of newspapers for a particular city, and the dates of publication, see:
A Reference Guide to Afro-American Publications and Editors, 1827-1946 by 
African-American Newspapers and Periodicals by This is a primary source-centric database that covers American history from the 19th century through the early 20th century. Search for newspapers, eyewitness accounts, and more for events such as the Civil War, women's suffrage, and the abolitionist movement. It also contains a collection of African American newspapers from 1827-1909
Search and display African American newspapers culled from the Access World News database from NewsBank. Exact dates of coverage vary by title, with most ca. 2000 - present.
This primary source collection of 1920s newspapers features significant publications of the Ku Klux Klan along with the Catholic, Jewish, and African-American newspapers opposing the Klan and white nationalism in general. For CMU users, the corpus is fully downloadable for text- and datamining purposes. Contact a CMU librarian to access the corpus in this way.