I'm not sure how safe your computer is. Is it your own personal computer, or do you share it? How do you keep him from accessing what you do? Give us a bit more info and we may be able to help you work out a safe way to do it, especially if you think he'll cause any trouble if he finds out that you are doing it behind his back or something lame like that.

You could keep a dream journal in a book he wouldn't be likely to read. Like maybe a textbook? Something that it wouldn't look weird that you had it in your room or if he ever caught you reading it, though.