Recently (today) I heard a clicking sound coming from the case and a short moment after that Windows showed a message that HDD could be failing and I should backup my data immediately. Ran "Check disk" and clicking stopped, but HD Tune Pro still shows bad things (at least from what I understand): Is it dead and I should start looking for a replacement or something can be done about this?
I would say i'm i the same position with the difference that i hear the clicking noise from time to time and windows freezes for some time but after that just continues working properly. I got concerned just like you are and as it can crash every moment i use it just for the operating system and keep nothing valuable on it. My opinion is if it still has guarantee, use it. If not buy new one. By that time just leave no valuable data on in.
That's called the "click of death". Your hard drive is literally on its last legs and at this point it may be too late to save all of your data. You could very well have many bad sectors on your disk and corrupt data, if your drive is able to read it at all. You have to replace it. That's it. If you want to try to get a little more life out of the drive (as in minutes or hours) to backup data, you can try turning your hard drive upside down. There are some Linux imaging programs you can use to read your disk backward and grab data off of it so it will not fail when it hits a bad sector.