I have a Asrock G31M-S system and when i plugged power the PC power button "LED" is Blinking, but the cpu cooler fans are dead, screen dead.
First of all, check if there are any inflated capacitors, if there is even one- ditch MBO, other capacitors will soon follow. If no inflated capacitors proceed with these diagnostic steps: 1/ disconnect all peripherials (DVD, HDD, card reader...), leave only MBO, PSU and RAM (and CPU of course); if nothing happens 2/remove CMOS battery (CR2032) for 20-30 sec with power cord disconected from computer, that will reset BIOS; 3/ take out RAM sticks, then, if you have more than one stick, try with each stick separately (it might be faulty RAM); 4/power on MBO without any RAM-if speaker is present it should give BEEP signals; 5/ try with different PSU, many power supplies are faulty even when they do give power to MBO.
If nothing of above does not help than you have faulty MBO, because CPUs are so rare to fail that in my 10 years of experience in PC service I had 3-4 fried CPU-s, and even that was because costumer tried to clean PC, removed CPU cooler and than incorrectly placed it back or didn't put thermal paste afterwards. 
.cap plague was real.
https://pcdied.com/CAP_PLAGUE/PLAGUE.htm...
2006-8 mother board.
the manual is here at makers site
https://download.asrock.com/Manual/G31M-...
that is correct PSU have many failure modes
dead or weak rail DC.
noise (ripple spec) on rails due to bad caps.
power on logic dead.
power OK logic dead.
overheats fast, gee 13 years of nasty lint inside packing up fan and its fan dead too.
not told that. OP
I load bank test my PSU and scope them for noise under full load, this is what works best but my shop is full of top instruments.
this mobo does have a beeper for sure if not lost seen here.
this beeper is polarize and means dead if plugged in backwards
+ to + , and - to -
no other way works.
if is connected right and does not beep now
that means mobo is 100% dead, PSU dead.
if PSU fan is dead that means,
PSU is dead or mobo has shorted devices on t.
the PSU fan means 12vdc is dead (rail)
and that is caused by
fan PSU stalled in lint
PSU dead
or mobo has hard shorts, 100% power overload.
missing ram beeps or wrong ram.
CPU bent pins , no beeps are common, for this.
all evercons caps on this mobo are JUNK.
the all leak or are bad. well known.
this is your same mobo, see it"?
one mobo like this NIB sold on ebay for 99cents
I bid 0
the now 15 year old coin cell is no good.
not possible good that old 2 times live (2x7=14years) 7 EOL spec, so is step one.
and sure bad caps. sure.
15 year old PC and monitor and coin ?
the monitor is dead, CCFL 100% dead now.
they last 5 years online, used 10hr a day
and the backlamps are dead. like clock work
the PSU and mobo both have bad caps.
both that same age, as it would be untouched by humans for 15 years, now.
my guess find this cheap in a flea market for $1 as is.
and is dead. RIP junk, G31 has no support nor any future of any kind,
buy PC H77 up, and win (chip set 2010 up)
G31 is series 3 , and zero dot zero support.
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I have try many 1. cmos clear 2. dissassamble 3. bios chip change ..... but no use
so did this PC run for years and fail now, not told
history matters. or found it dead. or bought it dead.
your post is only a MOBO, A PC has many parts not just a DIY MOBO.
no PSU told no things connected or if GPU card is present at all.
No DDR memory told test them 1by1. max is 3.2GB(4G phy ) DDR2 667/800MHz.
Sata drives?
the problem is. age and bad parts of assembling a relic, PC.
G31 is old.(intel G31) July 2007 born.
no CPU told , many even g31 rated do not work here and must be on ASROC QVL list for CPU, or it will fail.
so you are playing with a 7 year old mobo.
and the coin cell is wrong
the LED blink means power failure do not hot swap anything inside the PC, with out first removing AC power cored to PSU
no PSU told and some this old are bad, bad CAPS inside
but none told. maker nor model or wattage told.
first test the mobo with no GPU cards, no PCI-e cards no sata drives, and just 1stick of ram.
forget booting hDd (for now or any dead PC )
learn to get BIOS working first. stripped down.
BIOS must work all pages, and that means new coin cell battery(CMOS). max spec life is 7 years, bingo..
jumpers on mobo not set wrong.
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