To check overheating, use SpeedFan or any alternative.
http://www.almico.com/sfdownload.php
Explore this software without changing any settings and see HDD section. There you can see HDD maximum temperature, to check if it overheats or not.
If you have important data, then its better to seek professional help e.g. HP. Data recovery from failing or even failed HDD is possible mostly, provided there is no major physical damage.
If you want to DIY, Try HDD recovery softwares. They usually copy data in slow speeds.
Remember to copy data, folder by folder, not exceeding 500MBs in one go and skip the files that give errors. Moving is actually 'copy+delete old files' which doubles the load on HDD.
Another silly trick that mostly works for me. Use a basic class 2 or class 4 memory card from mobile or digital camera. They copy data very slowly (2-4 MB/s) and slow speed copying increases success rate on weak HDD.
Keep laptop on flat horizontal table during the process.