You are quite late in the game for salary negotiation in my opinion. If you accepted their offer, even verbally, they set aside certain amount of funds to sustain your existence in the company. Now, on the 11th hour, you come back with a counter offer, you had better have a very good leverage that makes them not being able to live without you. Since you mentioned first job change, I am assuming you are in the junior stage of your career. So, this company not being able live without you is not a likely scenario. So, tread the waters carefully at this stage.
Having said all this, you can go back to them, before handing them the background check forms and say something like:
After our last talk when we discussed my compensation, I did a little bit more investigation and found out that, this salary (or hourly rate
if you are being hired on contract basis) is significantly lower than
the industry average in this area. If we can not come to a mutually
agreeable compensation, I will not be able to work for your
And leave the ball in their court. If they come back with a better offer, or ask you how much you expect, you can tell them what you want, in a tactful way, by supporting it with data obtained from a reputable source (glassdoor.com for instance in US; since I don't know your location). But at this stage, be prepared to be dismissed from the hiring process and probably not being able to work for this company, as long as the HR and the hiring manager is at their same position.