I have HP Elitebook 8460p with Radeon GPU and without additional integrated intel graphics in CPU. Is it possible to run GPU mining on it?
You need a minimum of 2GB VRAM to mine ether. Since the laptop you mention has only 1GB of VRAM in the Radeon, then no, you cannot mine ether. As for other cryptocurrencies, you will get abysmal hash rates (speed at which you mine) so it won't be worth it.
You'd need a beefier GPU for mining to be worth your while. Here's an example of ether mining on a laptop: https://www.youtube.com/watch?v=n8OgNTOsRfQ
I recommend trying: BitMinter.
"If you just want to check that everything is working on your computer or what hash rate it will run at"
It depends on your goals.
In my opinion, that sort of "rig" isn't going to create financial success but you can run the software to learn how things work.
You can also get a feel for what winning would look like by mining on testnet where it's less competitive. In my experience, I've always been able to pick up a few testnet coins by using just Geth or Mist and letting a laptop CPU mine overnight.