I'm curious. Is this a lease? Are renting your computer out by the hour? You said you need to capture the time spent on the internet. If it's only time, the computer has a clock on it. If you are trying filter content that is different, i.e., parental control. It would also be unethical to bill them for web content you do not own. So why not just set a timer and forget about redirection? Why this redirection? If only just want them to keep going back to your page, just so you could get hits , oh, there are ways to trap a person to do that, but you wont get that from me because I despise that unethical way of monetary gain and its a very irritating experience to the user. Such abuse may be considered fraud and and person who does this intentionally may wind up in jail if reported. I'm not saying your're doing this, but you need to provide more details to why this is so necessary.
then again, is this a proxy? Proxy can have many legitimate uses that fit your description?