I think you're <i>describing</i> <a href="http://books.google.com.sg/books/about/Algebraic_Topology.html?id=h-wc3TnZMCcC&redir_esc=y">Spanier</a>.

Everyone I know who has seriously studied from Spanier swears by it- it's an absolute classic. The approach is exactly as you describe- algebraic topology for grown-ups. The treatment of homological algebra in it is extremely nice, and quite sophisticated.

A second, quite brilliant book along the same lines is <a href="http://www.amazon.co.jp/Introduction-Algebraic-Topology-Graduate-Mathematics/dp/0387966781/ref=sr_1_1?s=english-books&ie=UTF8&qid=1373292400&sr=1-1&keywords=algebraic+topology+rotman">Rotman</a>. It's more geometric than Spanier, for those who like such things, and find it easier to read (although that's a matter of taste of course). Again, the treatment is unembarrassed to employ nontrivial homological algebra and category theory, in a good way.