Usually I check what's running in task manager, to see if there's anything using an unusual amount of CPU or memory. If nothing shows there, it gets harder to diagnose as it could be a hardware issue or malware/viruses.
If it's an older computer with a hard drive, the usual suspect for me is hard drive wear or slowly failing drive with bad sectors. Usually I'll run a bootable drive scanning utility (I forget what's on UBCD... also need to get me a copy of SpinRite) and check the health of the drive. Replacing an old drive is the easiest solution if it's a failing drive. If it's an SSD, I look to see if the CPU is throttling because of heat (check temps using HWMonitor and/or Core Temp). Running a stress test (but not necessarily a "torture test") using a benchmarking tool like Aida64 while monitoring the temps is a good way to initially rule out thermal throttling.
If it's running hot - you can look up what's "normal" for your particular CPU - usually you don't want over 70C to 85C* as an upper limit, depending on whether it's AMD or Intel, but there's various opinions on what's okay for the long term reliability of a chip. Obviously the colder the better. The processor will normally trigger a shutdown of the computer if the temperature is running too hot even after throttling. Cheapest fix for a PC that is throttling is to clean out the case and fans with compressed air, if dust is the culprit. Desktop computers can suck in a ton of dust, practically acting as air filters. It can get nasty in there.
Sometimes a slow PC is the result of Windows getting bogged down (that's the technical term
) by a lot of running software and bloat - more common with older versions of Windows and PCs - and not necessarily a specific program. In that case, a quick wipe/reformat and reinstall of the OS does the trick. I still shudder recalling the amount of defragmentation I had to do on old hard drives in the Windows 95/98 days.
You can also try one of the reset or refresh options in recovery if it's a new-ish version of Windows. Clean Windows is happy Windows.
If none of the above works, it's time to start ruling out other individual components and issues. Bad drivers, corrupted OS (though you'll get other errors and not just slowdowns if that's the case), poorly optimized services/software running in the background, etc. On the hardware side, you have to look at video card temps if the slowdowns is in gaming or other GPU intensive programs, possibly check for firmware updates for the motherboard if there's a known issue causing the mobo to be flaky... there are a ton of potential causes, but usually other than thermal throttling, a PC won't only slow down if there's hardware failing, you'll get blue screens of death and other symptomatic errors. Googling the manufacturer or model of the PC and specific parts will often turn up forums like this where you can see if other users have experienced similar issues for problematic parts. Oftentimes certain models of PCs - and especially laptops, in regards to heat dissipation - are just poorly designed or have faulty parts with bizarre issues that would be hard to isolate yourself.
There are a lot of other diagnostics tools and steps I'm leaving out, but the main thing is you want someone who is good at this to look at the computer, if possible, because fixing computers can be a royal pain in the butt. It doesn't hurt to try and narrow down what the issue is yourself, though, even if you aren't a computer person. Just don't use any of those magic programs that claim to fix your pc with a simple click. Those often can be placebo apps at best and malware at worst.
One other possibility if nothing appears to be wrong - and especially if the slowdown was over time and/or after major software upgrades and installs - is just insufficient hardware to handle whatever load the user is putting on it. Adding/upgrading the RAM, putting the OS on an SSD, or just buying a new, higher spec computer is often the least hassle, even if it's not the cheapest.
* sensors can fail or be inaccurate, so there's another potential issue to rule out
Luckily that's usually not the primary issue.