How do I properly credit authors of software released under the MIT license?

Question

I have modified the source code of an MIT licensed project, and I have added new classes to it as well. Please correct me if I am wrong, but I think that it is legal to add my copyright notice above the license and remove the other one. But how should I attribute the contribution of the previous authors? Should I use a separate file? There are also some HTML files with no license or copyright notice in them which I also modified. Do I have to handle them differently?

My question is different than this question in that I have also modified some of the files of the project I am extending.

Update

Although the suggestion to remove the copyright notice sounds weird, what I had in mind when I first posted this, is that the authors should not be responsible if I add something malicious to their code. This should not be a problem since MIT license includes a disclaimer.