Anything on the screen during this?
If nothing on the screen then it is usually time to check the RAM is in properly and also check the graphics card is in properly, if you have one that is. Dead RAM is a thing but it is very rare, dead graphics cards are rather more common.
What is the PC? Said alarm noises are often error codes (so called beep codes)
http://www.computerhope.com/beep.htm for a basic overview, whatever makes your BIOS or the motherboard manual should tell you more.
I have also had them be heat alarms in computers which might cover the next thing.
The click noise could be a whole bunch of things, some not good and others fine to ignore or trivial to sort (could be a lead clipping the blades of the fan), but without hearing it or looking at it I have nothing really to add.
"leaving it off for a while, it boots up into Windows and on the signage screen just fine"
This could be many things, as mentioned above it could be overheating for some reason and leaving it off gives it chance to cool back down. I have seen everything from fans falling off the mounts, clogged with dust, simple overheating (usually on small machines) and usually fairly easily remedied when you take the case side off and see what is happening. Very rarely it will be something non obvious and closer to an actual fault in the design, usually another high performance chip (depending upon the age of the machine then the northbridge, or the chip between the RAM and the CPU, but modern CPUs stuff that inside the CPU as well) but go with the basics (fan not mounted properly, dust, not plugged in properly, any head sensors not where they should be...)
