"Is it as simple as buying a decent wireless router and installing it?"
Yes.
"Do I have to get in touch with my cable company (Charter) for any reason? "
No.
When you configure the wireless router, there will be a setting for your type of service: PPPoE, Cable Modem, VLAN, etc... Just select the appopriate type, in this case "cable modem" and you should be set. There might be some additional data to enter, but that is minimal, just set your computers for DHCP and you should be set.
You know my email addy, feel free to shoot me any questions you have.