If I understand correctly you want to start with your inbox? To skip Windows Live Home when you sign in, change your settings as follows:
- go to hotmail.com and sign in
- click your name on top right
- choose options in the dropdown
- below header Home, click "Customize Windows Live Home"
- select "show me my inbox when I sign in"
- click save and you are done, next time when you log in on hotmail.com you will be led to your inbox directly