I will tell you about what are the things you should look for and have to be careful while buying a second hand mobile.
First thing nobody will sell his/her mobile without any reason. So you should know about why he is selling his mobile, is it there any fault in the mobile, he will never tell you about the fault. So you can tell to him that before buying the mobile I have to use this mobile for 1 week, then I can buy this. Otherwise if he refuse for that then dont buy.
Second the mobile should have new features such as minimum 2 gb ram and quad core processor. Look for the camera it should be more than 5 megapixel if you need a good quality camera.
And also look that battery is removable or non removable. If it is removable, check out it copper pins, is it looking fine or not.
Do multi tasking such as play games, use whatsapp and other apps in a one time, then check the mobile is getting slow or doing perfectly fine.