What is the best way to tell my dream company that I'm interested, just not right now?
Be honest with them. Say you are interested in working from them but are not interested in a new role at this time. Many large tech companies (e.g. Amazon, Google, Microsoft) have annual intakes and will just add you to the list for next year.
Alternatively, if you are talking to a hiring manager instead of HR, keep that connection open. Have the occasional coffee with him/her, chat with him/her at user groups or conferences, wish him/her happy birthday on social networks and so on.
In the meantime, make sure you keep up with the technologies and trends your future employer will want. Keep your skills current and demonstrable. Make sure you are familiar with new products and their intended use. Focus on the business side, too, not just the technical. For example, what companies have they recently acquired? Why?
Lastly, remember that hiring at large tech companies is usually done on a schedule that benefits the tech company, not you. You may need to leave your existing work in the middle of an important project. I am not saying never take on a new project or important work. I am saying you may not be able to keep everyone happy.