I've just looked up the manual of your system, and it has a dsub connection, not an hdmi one. Have you installed a discrete gpu? If not, try changing the input mode on your monitor whilst the pc is on.
Have you tried a different monitor?
Or, the computer may be turning on but not posting, hence no output to monitor. Could be a memory issue. Try removing / swapping memory sticks if possible. Could also be a psu issue, these emachines are always issued with pants psus.
Does the machine sound like it normally does when booting up? Is the hard disk grumbling away? If not, then the machine is likely not posting.