<p>Once I have my renamed files I need to add them to my project's wiki page. This is a fairly repetitive manual task, so I guess I could script it but I don't know where to start.</p>

<p>The process is:</p>

<pre><code>Got to appropriate page on the wiki

for each team member (DeveloperA, DeveloperB, DeveloperC)
{
    for each of two files ('*_current.jpg', '*_lastweek.jpg')
    {
        Select 'Attach' link on page
        Select the 'manage' link next to the file to be updated
        Click 'Browse' button
        Browse to the relevant file (which has the same name as the previous version)
        Click 'Upload file' button
    }
}
</code></pre>

<p>Not necessarily looking for the full solution as I'd like to give it a go myself.</p>

<p>Where to begin? What language could I use to do this and how difficult would it be?</p>

If you're writing in C#, the WebClient classes might be a good place to start.  I bet people could give more specific advice if you mentioned which wiki platform you are using, and whether it requires authentication, though.
I'd probably start by downloading fiddler and watching the http requests from doing it manually.  Then you could use some simple scripts and regexes to build your http requests for automating the process.
Of course, if your wildly lucky, your wiki would have a backend simple enough that you could just plug them into its db directly. :)