Sorry this reply is kind of late
I actually considered an Alkaid costume for my friend (since she didn't know much about .hack) and thought it through a little.
I recommend Simplicity 3675 for the pants (I used it for Sakaki), and lots of interfacing for all of the stiffer parts of the cosplay. The wig looks a little hard to me, but if you're familiar with wig-dying (I use the sharpie method) then it's possible to the gradient on her hair with a few different colors (and some patience!)
I don't know if it's at all Joanne's stores, but mine had a sale on all furs as of today, and I'm halfway sure the sharpie method for dying wigs would work on the fur as well (I know it works on felt). You should be able to find the cord pretty easily, and the stripes could be done with bias tape or even ribbon (you could sew or use stitch-witch). And I'm pretty sure there are boots like Alkaid's available on cosworx.com.
I wish I could help with her shirt, and the only way I've known to put on tattooes is with lip liner, but it smears fairly easily.
Hope this helps!