It's difficult to know exactly what you mean by background noise without an example.  Can you post a video?

If you are just starting out, then it takes a while to refine muting.  The first thing to do is to maintain contact with the strings.  Do not ever release contain with the strings.  As soon as you are done playing a note either playing the next note, or at least release enough pressure to stop the note sounding, but maintain contact in order to keep the string from ringing.  When you get ready to move to the next higher string you can mute the lower string with the your fretting hand fingertip.  Also make sure your picking is as minimal & economical as possible to avoid striking strings unnecessarily.

If you are using a high gain tone, then there is likely going to be some level of extra sound.  But usually if you are playing a fast enough line, then the notes happen too quickly to notice.  Sometimes it happens when you move to the next string overcross strings.  Remember to maintain some kind of contact with the strings with your fretting fingers.  That's also when muting with your picking hand becomes useful.  Always keep your palm in contact with the strings.

If you are using a clean tone then it's absolutely possible to reduce unwanted sound with the right muting & picking techniques.  As ever make sure your fretting fingers are always in contact with the strings, even when you aren't playing a note on that string. 

I have a whole tutorial aimed at muting at beginner, intermediate & advanced levels.

https://www.guitartricks.com/tutorial/2281

And these tutorials are aimed at playing fast lines on the top 2 strings with a gain tone. 

https://www.guitartricks.com/tutorial/928

https://www.guitartricks.com/tutorial/929

I focus on muting in the 3rd lesson.

https://www.guitartricks.com/lesson/14295

Hope that helps!