My domain name (peteg9699.com) is expiring June 1. How do I go about renewing the name? I received a bill from Libertyname.com or something along those lines, but I do not remember registering the name with them, I thought I had registered it through hostony.com. Any help would be greatly appreciated.

If your domain name is registered not with our company you need to contact your current registrar ot know how to get a renew.

I registered the domain name with your company when I signed up for the hosting.

Use following steps to renew your domain.
Go to http://hostony.com/pay/domainname.html
pay for domain name.
Open new trouble ticket with your domain name and order number.
Your domain will be renewed.

How do I open the trouble ticket to let you know I paid for the renewal of my domain name?

Just go to http://support.hostony.com and you'll see