My wife's iPhone has iOS 5 and hence iMessage installed. I would like to be able to chat with her using my Windows computer. What options are available for this? Is there a Windows client or any web site that can provide a bridge?
As of May 2016 the only official iMessage clients are for iOS and OS X. Unfortunately you're out of luck using it on Windows.
It's very unlikely that any iMessage clients for non-Apple platforms will be released in the future. It's a major selling point for Apple's platforms, and therefore not in Apple's interest to make a Windows (or Android, Linux, etc.) version. Additionally, it has some fairly complex encryption and security protocols, which likely make it very difficult if not impossible for an unofficial client to be developed.
As noted above, there currently exists no iMessage client for Windows. However, your question seems broader than that: your wife has an iPhone, you have a Windows computer, and you want to be able to communicate with each other.
There exist a number of options that could allow you to communicate.
If you wish to literally chat, you could use the embedded phone call feature in gmail (requires plugin) on your computer to call your wife's iPhone.
If you wish to type and have messages appear on each other's screen, both Windows computers and iPhones come with e-mail clients.
If you wish to send messages more optimized for shorter communication, you could create a Google Voice account exchange send SMS messages between Google's free interface and your wife's phone. To avoid SMS charges on your wife's phone, she'd need to install the Google Voice app and do the messaging through the app.
If there are particular features of iMessage you wish to exploit that these methods do not allow, please clarify your question to highlight the features you are missing here.
No, there is no iMessage client available for a Windows PC.
iMessage is based on a propietary, binary and encrypted protocol (source Wikipedia) with a client side device certificate, which makes it unlikely to be reverse engineered and implemented in third party apps any time soon (although attempts are being made).
If your device is jailbroken, you can purchase the Remote Messages tweak from Cydia ($3.99, currently) and use iMessage via a web interface from anywhere (PC included).