I see below points here, which needs to be fixed:
Once you deploy your application in tomcat webapp directory, the URL should be something like
http://localhost:8080/<web-app-name>/<your servlet>
Alternatively you could add a welcome page and check if it is working. For that you need to add welcome page setting in web-xml. See This post for a clue.
If this is not working, please share your tomcat directory tree with your war deployed; and the war file directory structure
Note: you should be able to expand your war using a zip utility.