So I've got this picture that I've cut out and removed the background from and I've fixed it up to as close as I can get to the picture itself and now I'm left with a pretty bad looking outline with pixels that I want to blur into the new black background, such as to soften the edges of the picture perhaps? (New to photoshop, I honestly can't be sure what I'm saying is 100% true to what I'm actually trying to achieve) Any help with this is appreciated.
This depends upon how you cut out the image. If you erased it then you can ctrl/cmd-click on the layer's thumbnail, which will select all areas that have any pixels. Use the feather option in the selection menu under modify to soften the edge of the selection. Then you can click on the mask icon in the layers panel to apply this selection to the layer. You can then use different blurring tools to adjust the feathering.
If you used a mask to cut out the image, there are the mask refine tools to help adjust the edges of the mask.