Is it possible to follow someone on Twitter without them knowing?
If you use a feed reader such as Google Reader, you can subscribe to the RSS feed of the twitter user you want to "follow". Basically, it'll be like visiting their Twitter page, but you won't show up as a "follower" on Twitter.
Put them on a private list. You can then just visit the list to see their tweets. You might want to also add the people you're actually following to that list and change your bookmark to point to it instead of Twitter's home page.
If the account is unprotected and you're using Firefox, there's a plugin called TweetStalk that lets you stalk users instead of following them.
You would have to do it manually (by going to their page and seeing what they post). You can't add their content to your homepage unless you follow them.
You could write an application that pulls your homepage content and mixes in their timeline content. You could also use a RSS reader to pull in multiple feeds (your homepage and their stream) into a single location.
Note: They will only be notified if they have notifications turned on, or they constantly monitor their list of followers to see if it is changing.
Jared Hardley's suggestion (RSS) is the way to go (for non-protected accounts). If you want the person's tweets plus any replies (which you can get in RSS via a search), you can use this Yahoo Pipe I created that mixes both things up: http://pipes.yahoo.com/pipes/pipe.info?_id=a9379d57504827fc594e71daeb54dd60