Your @ level 7. You want a gaussian blur, yet you dont know how to apply it. This is the first time I've used the expression, but, OMFG. Uhh... if you're using photoshop it's under Filter>Blur>Gaussian blur. Good luck getting anything posted on PJ using that method though...

I want to give depth to the background and give the character role

Why would you wnat to blur pixel art?

I want to give it a Gaussian blur ... I honestly do not know how.

'The Pink looks like its blurred on the edges, i think you over-used the AA.