What is the ruling on registering on the website (www.goodreads.com), which is concerned with books, and is that considered showing off (riya') and contrary to sincerity (ikhlas) in seeking knowledge?

There is no harm in registering on this website and benefiting from its content. This is not considered showing off (riya') which negates sincerity in seeking knowledge. What matters is the intention. So, be sincere in your intention when seeking knowledge, and Allah will teach you and bless you.