The Philips 24" LED TV is a monitor that also has a tv tuner. It's likely already 1080p. Having a tv tuner doesn't make it a bad monitor. A gaming monitor at 1080p might have slighly less lag, or might have less ghosting, but most likely it's the same. (Typed on a 1920 x 1200 24" lcd TV that I use as a monitor). Keep your current TV as monitor unless it is "720p". Google your TV's model number to find if it is 1080p (good) or 720p (bad).
If you mainly play games a stronger Video card (also called GPU) will make the most difference. To get a good/valid recommendation you need to give your current PC specs including CPU, power supply size, current video solution, etc.) Googling your PC model number will help. Then tell us the case size for your PC. Some really slim cases cannot take a normal size video card, they need a "low-profile" card. Unless people know that they will make bad recommendations. Also saying how much you want to spend gets people in the right ballpark.