Sure it will.  Just follow the setup instructions for the PC.   Internet service to the cable or dsl modem is what you pay for -  ISP's like Charter don't care about which individual PC (or multiple PC's) you have connected to the modem.  

Or did you have a more specific question,  like connecting both your old PC and the new one to the net at the same time?  That depends on what connections your modem and/or router have.  For example,  If you have only the basic cable modem with one ethernet connection , but you need 2 connections, you'd need to add a router, or replace the modem with a combination modem/router unit with multiple connections.