Here's what I would recommend ...
(1) If you're using any power-strips, replace them if possible. They wear out eventually.
(2) Before turning on the computer, turn on some lights in the same room and go over to the breaker box. If your breakers aren't labled, you'll have to go reset them one by one until you (or somebody else) tells you that the lights went off in that room. Once you've isolated the breaker, look at it. It should tell you how many AMPs (i.e. 15, 20, or 30) its good for before it trips. Consider asking an electrician to replace that breaker.
(3) Homes and apartments built around WWII or prior are increasingly becoming inefficient when it comes to electricity. Often times, the wiring has degraded so much that the entire place is no longer grounded, and if you ever intended on cashing in on the guarantees of some surge protectors and UPS units, they explicitly state grounding is a must. And when your house/apartment isn't well grounded, your electronics become more susceptible to what is known as "dirty power" ... http://www.electriciantalk.com/f2/what- ... wer-16131/
... and they can literally eat computer PSUs alive - even the best top of the line switching PSUs from Seasonic and FSP. Keep in mind that it's normal for switching PSUs to make a sound every once in a while like just earned a free credit on a pinball machine. Just not as loud.
A faint "KLOCK" sound is perfectly normal. But if you hear it more often than not, could be a sign the PSU is taxing itself. Of course, its expensive to have one's house or apartment rewired (or getting the land lord/superintendent to do it), and that's where a decent UPS comes in. Even the cheapest ones on the market that'll maybe give you 30 mins to an hour of power if a storm or accident takes out a uility pole have good tolerance for dirty power and automatically condition the output signal with a clean sine wave that computer PSUs love. Plus, its an overall good idea since that last thing you wanna end up doing is re-installing your OS after a sudden power loss boarked your system. So a UPS gives you a good grace period for getting documents saved and everything shut down normally.
(4) Easiest one that doesn't cost spending money you may not have: Re-locate the computer!
Lights shouldn't flicker or go dim when you power up a computer unless there's a short somewhere in the in-house wiring or the breaker to that room is being stressed and about to trip. Sadly, most homes and apartments will slog the utility room where the washer and drier is on a 30+ AMP breaker ... and leave the living room, kitchen, and bedrooms on 15 AMPs or less; 20 AMPs if your lucky. My kitchen is on a 20 AMP -- can't run a coffee pot, the toaster, or the microwave at the same time. Good thing the stove is gas!