Have you defined your new 2-monitor setup (one for the PC monitor itself, and a second for the HDTV) as an "extended desktop"? Then one of the monitors becomes #1 and the second becomes #2. Catalyst Control Center can be used to configure your dual-monitor setup, and which is "primary" (i.e. #1), etc.
With a 2-monitor setup you can run "extended desktop" (i.e. double-width desktop, and windows can be dragged around from one monitor to the other) or "clone mode" (duplicate output from monitor #1 sent simultaneously to monitor #2). This is up to you.
In the absence of proper dual-monitor configuration setup , I'm not surprised there would be at least some confusion (e.g. if the HDMI cable is plugged in to PC and HDTV, but the HDTV is powered off).
What have you done so far (if anything) in the way of additional CCC "dual-monitor" configuration, in support of adding HDMI-to-HDTV output from your HD5450?