I'm trying to make a new "serious" character, as in immersion serious. I don't know what to name it though, so can anyone link me to a website where I can find a "fitting" name to my character according to race?

Ever heard of google? Type skyrim name generator or something.

Ted's roman name generator is great for finding out your name in Latin.

This should help:
http://tamriel-rebuilt.org/index.php?p=modding_data%2Fngen&race=IMPERIAL&gender=male&count=10

How about you give us some corner stones of your new character?

What class shall it be? What skills shall it posses? Its gender would be helpful, too...