Does the Biclique attack on AES pose a credible risk to its security? I have heard a lot about the biclique cryptanalysis research on AES, which as far as I know is the closest anyone has got to breaking AES.

Exactly how close did they get? Does this attack propose a credible risk to my using AES today? Exactly how hard would it be to extend their work to fully break AES?
Simply put no.

As per the abstract, those attacks take at most 4 bits off the key space, this still results minimally in 124 bits of security. Put another way, to use these attacks you would need to expend effort roughly proportional to brute forcing an AES key where you already knew four bits of the key and this would take approximately $2^{124}$ operations. 

For reference, the standard rule of thumb on security is it must provide about 80 bits currently. Of course, this is to be secure now and not 100 years from now and and something that provides 80 bites of security now could have those eroded later. However 124 is still more than enough of a safety margin.