I've had an Asus G60J notebook for a couple years now and it has worked with excellent performance up till about 5 months ago. Now anytime I play games that shouldn't be hard to run at all, it over heats in about 15 mins. I don't even have enough time to set my character up to even play the game. When the computer is idling, the CPU runs at 53 Celsius (127F) and the GPU runs around 55 Celsius (131F). As soon as I start a game, the CPU rises to around 90 Celsius and the GPU rises to around 90 as well. The temperatures continue to rise until the whole thing comes to a stop. I have tried many things to fix it including changing the thermal paste (Arctic silver 5). This lowered the temperature just a small amount. The fan is a bit dusty and I haven't picked up some computer duster to fix that yet, but it seems that there is a bigger issue than dust. Should I take it somewhere to get it looked at and fixed?
^^^^ You need to blow the dust out of the heatsinks and fans. That is probably your problem. Also, make sure the air intakes and exhausts aren't being blocked by anything when you are using the laptop (e.g. if you set the laptop on a thick carpet or squishy surface, the bottom vents may be blocked by the carpet/squishy surface). Rigid surfaces or laptop cooling pads help prevent such blockage.
Your entire cooling system if clogged. I had this same issue happen to me on an older laptop. Long story short, I had to physically take the laptop apart to the point where I had access to the fans and could pull the massive clumps of dirt-like dust out of the path of airflow. You could pay someone to fix it for you, but they'll just do exactly what I am telling you to do and there are usually guides (google it) on how to disassemble your laptop and not destroy anything in the process.
Either way, if you don't fix your airflow problem (and that's what this is) you're going to melt your computer. My advice, don't game until you take it apart and clean it.
Edit: When you take it apart you might have to reapply the thermal paste again; make sure to clean it (the heatsinks, CPU, GPU) and apply the paste correctly when you do.