I got the same processor!
I love it!
I over clocked it to 2.8 GHz :D, so I may can help you.
I got 2 GB ram.
You don't have to worry under 50 Celsius I would say. I have bought a new cooler, tho.
"So is it appropriate for me to do?"
If you own a C2D E6300 you MUST overclock it.
I have done this about 2 years ago and I haven't done with it anything since that so I forgot this and that.
How you can overclock it is highly depends on your mainboard and your ram too.
When you overclok you must raise the voltage a bit(with E6300 it's not much), and set the multiplier of the ram to fit the new core speed, maybe you have to set the ram frequency too.
I know I haven't provided much knowledge but this may help:
http://www.overclock.net/You have to google your ram, what properties they suggest for it(Mine is 4-4-4-12, Geil Ultra 800 MHz) and set it to that. And when you overclocking you have to keep the ram frequency within a small range with modifying the multipliers.(For me it's 400 MHz but it counts double for me(I don't really understand this part.)
You don't have to worry, intel processors won't burn in your mainboard. If you don't do something really serious you will just have to reboot and set the things back.
Don't rely on programs that do this things for you after startup.
Just set everything with hand in BIOS.First just give it a small boos. Maybe 1,9 GHz or 2.0 GHz. Don't even bother with other things. Try it. If there is no beeb you are okay :)
You must get CPUZ and some stressing programme(I forgot what I have used) to test the stability of your new setup. Then reboot -> BIOS -> some more MHz, voltage(very little) -> test again. You will have to do some "jumps" though because of the ram frequency so that you can keep it in the small range with changing the multipliers.
For example you want 400 for the ram. As you changing the core speed it will raise too so if it is like 420 change the multiplier so you will have 380. But 380 is low a bit so this is the jump, you boost the core speed a bit more so it will be near 400(the ram).
My mainboard count me the ram frequency, some not, so you may have trouble counting that yourself(I don't know the formulas).