I just renewed my expiring card, by getting to that first link Andre supplied (above your post)...but I'd found that page on my own. I didn't "delete" anything first.
Under the "Updating the credit card used for a subscription" (since that's what I have) section on that page, I followed Step 1, 2, but NOT Step 3 (which you'd use, I guess, ONLY if you want to SWITCH your pymt method; they should make THAT clearer, too, by adding an "OR" into the sentence)....then did Step 4. That takes you to an outside page to verify your credit card info with your banking institution. However, you have to know your "password" for that card account, first! (had to then go searching for that, adding even more time spent)
Now I'm waiting for the verification of my account/pymnt changes....hope that doesn't take too long, adding yet more stress to this process!
But I agree -- I've never seen such a confusing morass of instructions anywhere simply to update an expiry date on a credit card! Had to read everything over & over again to figure it out, then ask my spouse for his "take," wasting MINE AND HIS valuable time. It's normally such a quick-click process elsewhere.
What was Skype thinking???! They really ought to re-jig this method. 
