I'd go with the comments from flowerybeanbag
and assume anything you do can and will be logged. It's easier to assume that, and is not unreasonable - anyone using an 'office' mail address who thinks it is 'their right' is sadly misguided... better to use computer access at the library, or on an iPhone (in lunch break) or anywhere but the office, if you want to avoid any chance of claims of "misuse of equipment" that might go towards disciplinary procedures.
In networking terms, all packets to/from an individual PC can be logged and timestamped, so a profile of activity can be created (and even though content might be encrypted, the times of activity would show how many minutes per hour were spent connecting to outside site{s}).
In theory, one could view a site and just minimise the browser, but it could reload the page every 5 minutes (try www.NewsNow.co.uk for an example) and accidentally give the impression that the user is checking the web regularly... It is not always clear that a site regularly refreshes the page, and it's possible for a background page (similar way as pop-under adverts) to be created that refresh every few minutes, unknown to user.
Similarly, there'd be regular activity if you have something like the little 'Gmail Notifier' program (that connects to the Google Mail service and pops up when new mail has arrived).
For LIZS - better to have a second machine of your own connecting to the router so your second machine isn't in any way connected to the company traffic.
However, there are some services offered which are intended to only connect you to your company network, whatever the PCs, but it would surely be clear that that was what it was provided for (and presumably it would be on a separate and funded phone line - www.Opal.co.uk offers VPN service for companies/ small business, for example - there would be absolutely no way "around it" in that situation).