Monitor No Signal or Going to Sleep When Connected To Laptop?

Yep that is what we need.
Here is the driver page on Asus website for your laptop.
Notebooks & Ultrabooks - U45Jc - ASUS

You use the Windows Bios Utility to upgrade the bios. I seriously doubt that is the issue. Looks to me the bios was updated for those who downgraded to Windows 8. I have asked an laptop expert who knows more than I do about laptops to take a look. I suspect the Video chip could be overheating and needs the dust and fluff that may have built up in the machines fans removed and possibly some new heat sink compound on the cooler heatsink. Could be a very easy job or a nightmare depending on how the machine comes apart. May not be something you want to do yourself as it can be easily damaged.

Just my 2 cents worth it seems you are trying to drive a fairly high res monitor from a GPU (and CPU) that are having a job keeping the device functioning.

I stand to be corrected but did you say it was an old monitor and could I suggest you get one of the low powered LED screens - and not overly large?

As for cleaning the air cooling system I always use my laptops on a hard surface not on ones lap or any surface likely to cause a lowered air intake.
Have a read of my ditty here

CLEANINGLAPTOPS

What you can do is to check the air intake aperture and the"rad" and to do that shine a very bright light into the fan apertureand look through from the outlet grill
1. You will see if there is a lot of dust in the fan intake and fan blades and2. you will see if the "rad" is blocked in any way when you lookthrough from the outside.

Now to clean you can blow the dust out using compressed air but first makesure you insert something into the fan blades before you do - to stop the fanfrom spinningas spinning the fan withair will cause it to spin at much higher revs than it was designed for and canwreck the bearing.
The rad you can use those chenille wire play things to very gently clean therad from outside the machine and again blow the dust out. It is fiddly buteffective.

Having net connect problems but will send a pic.

I meant to add a pic to show you what is involved - this example is very basic but is typical of cooling in a laptop. On my laptops on the outside I tape a piece of fairly fine material as a filter because the intake is very close even on a hard surface to dust / crud.

The first thing is I see the res is not having much of an issue on the machine.
Secondly if you follow my instructions that cleaning is very easy to do - did you by the way shine the light in the air intake and if so was there anything to see in the outlet grill? If not then that rad is blocked and like I described if you are very gentle with the chenille pipe cleaners you cannot do much damage.
I tell people that because it would really take excessive force to bend the rad fins - for example ramming a small screwdriver in there. It can be done also with a very thin strip of old credit card with no embossed numbers on it but take care not to snap it off when pushing through the rad fins. It is really common sense stuff.
The cost of a techie doing that would be very expensive as he would disassemble the machine and at rates like (out here) $85 per first consult and $75 per half hour after works out as much as the machine is worth alomost
The most important thing is to stop the fan from spinning too fast or at all when using a compressed air can.
First the light shining please.

Now without reading all the ay back what are the temps for the machine when running it?
Speccy will do Speccy - Download