Computer turns on but no signal in monitor

I believe your problem is that you bought an "AMD Athlon X4 860K" processor expecting it to have integrated graphics, however it does not. As you have listed your parts, there is no GPU in your part list.

If you would like a processor of the same level as the one you bought with an integrated GPU, I would check out the A10-7850.

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Edit:

As it has been stated, it is probably easier to buy a GPU at this point, so if you are looking for a good GPU for a decent price XFX Radeon R7 360 or an ASUS GTX 700 GeForce GTX 750 Ti as both of these GPUs are good, budget GPUs that could actually play newer games at medium setting while getting 60+ frames.

Also, I am inferring that this build will be used for gaming because the average user doesn't tend to build their own PC. However, if you plan on just using it as an office computer, like user34716 suggested below, you could purchase a low end GPU, like the GeForce GT 610 and be perfectly fine.

The problem you have is that your CPU does not have integrated graphics, and you don't have a graphics card. Rather than invest in a new, expensive processor and potentially risk buying a processor with an incompatible socket type, simply buy a graphics card. If you don't plan on gaming, you can pick up an adequate graphics card for quite cheap. A gt-610 or similar GPU should suffice. A cheap graphics card will have significantly more power than CPU integrated graphics.