New power supply

You replaced power supply and your made sure you plugged everything back into the motherboard snuggly and in it's proper place? If so it might be your motherboard is bad.

Make sure you have the new power supply turned on. Sometimes there's a O/I switch on the back of the power supply. Flip it and try to boot and see what happens. You might have another bad power supply.
If so your motherboard might be burning them out.

Try removing the power cord for 30 seconds and press and hold the power button. After 30 seconds put the power cord back in and press the power button to see what happens.

Do your fans work when you press the power button? Do you hear any fans running?

This is really poor practice and an excellent way to destroy a motherboard.
You should always have the board connectors plugged in before turning on the power supply, otherwise you may be blowing ICs left and right. It's perfectly OK to plug the supply into the wall or turn on its power switch with the board connectors in place. Also, nearly all modern boards have a set of pins that go to the front panel 'power' switch which does not switch AC but sends a logic signal to the power supply to turn the rest of the voltages on. Those pins have a 5V potential on them that is typically grounded to tell the supply to turn on.
If you're lucky, you still have an undamaged motherboard.

See if you plugged in the 4 pin power connector on the motherboard besides the main power connection on the power supply (usually black colored and its a plastic 4 pin connector)

Try another power supply if all else fails. If new power supply does not work with this unit then its a motherboard which is blown and motherboard needs replacing.

Hope this helps

Most likely scenario is either the power supply or motherboard died. Check to see if the led near the main psu plug to the motherboard is green. If it is red or not on then more than likely the psu died. If the light to the mb is green then check your RAM and video card as those components going bad or coming unseated would also prevent the system from booting. Once you find the defective part replace with a new one and you should be back up ans running. Good luck & I hope this helps.

I would suggest trying to replace the power supply. If that does not fix the issue, i would need to hear more about the computer.

Do you mean the little green LED light near where you plug the power cable into the back of the computer? If that's the blinking light then unfortunately, this means that power is getting to the power supply but it is not going from the Power Supply to the Motherboard/Devices. The power supply is broken and is not converting the power from the wall to the proper voltage for the Motherboard/Devices. You will have to replace the power supply.

Call HP, and I hope you are in warranty and get it replaced for free or you can order the exact same one from HP that originally comes with that model. 1-800-HP-INVENT for their tech support line.

it sounds like the hardrive you installed overloaded the system and cause a low power operation on the computer the motherboard may be bad because of this as well as anyparts that were connected to it i have had this happen to me before and well i know it sucks but your best bet is to just buy a new computer and make sure you power supply can handle extra parts...

I would unplug the power supply from the motherboard, wait a little while and then plug it back in.
Try to power everything back up and it should come up.

I am assuming you are getting a light on the motherboard that indicates power is being supplied?

Unplug computer, disconnect all usb/monitor/internet, open it, hold power button until green led on the inside on the motherboard turns off, pop out the BIOS battery (looks like a big watch battery and use a small flathead screwdriver to depress the spring lock on one side), plug in computer, hold power button for about 20 seconds (must reset BIOS), let up on power button, wait 10 seconds and push power button. The computer should turn on. Wait for it until activity seems stable (1 min. or so). Push powerbutton to shut off.

If this has worked you have proven that it's not the power supply (it turned on). If you purchased a non dell power supply, you might try this with the original one installed.

Go buy a new battery (CR2032 3 volt). Unplug computer, open it, hold power button until green led on the inside on the motherboard turns off, pop in new battery, plug in (you should notice the power button blinking amber again), hold power button until it stops blinking (about 20 sec....again must reset BIOS), wait 10 sec., push power button. the computer should turn on. Wait for it until activity seems stabe (1 min. or so). Push powerbutton to shut off.

Now, close case, plug in all usb/monitor/internet/etc. Power up computer. You'll get an internal floppy drive error. Push F2 (maybe F1) to get into the BIOS settings and go to "drives" and turn off the internal floppy. Save and exit. The computer should boot properly now.

This seems to be a big problem with the e510, my theory is that this one small battery is the root of most of these failures (as these computers usually fail after 1-2 years), not the power supply or the motherboard (heck, when you replace the motherboard it comes with a new one of these batteries...a several hundred dollar piece of hardware, with a $5 battery that will actually fix it).

Either the new power supply you purchased is not functioning correctly. But, assuming it is now that you are getting some form of power to your PC, it is very likely the motherboard. Whatever fried your old power supply (or it could have just been the power supply itself) most likely fried your motherboard in the process. Probably not the CPU since the MB can't even post.