I like the NFL draft.
It isn't perfect, but it is, in my humble opinion, infinitely superior to the draft system in the other major sports.
My only change might be the Brears predilection towards trading away valuable first-round picks, but that's not the league's fault.
I wouldn't change the draft as I like the way it is. What I would consider changing is how you can trade draft picks, perhaps something similar to the NBA where you cannot trade a team's own consecutive 1st round picks. So, for the Bears, they wouldn't have been allowed to trade all these 1st round picks lately, they would have to do every other year kind of deal. This way, you don't give up the farm for a young QB and then get him killed the next 4 years as you try and recoup picks. The team still pays a price, but you can also continue to strategically build around a franchise young player.
I also wouldn't mind seeing some kind of NBA style of Max Deal (salary and contract length). The Patrick Mahomes style of deal is only going to get worse and really hamstring an organization down the road. If not the NBA style of Max and Super Max, perhaps something like no play can make X% of the salary cap. This would help in the area of competitive balance.
Lastly, I would really like to see something done with regards to "Dead Money" but just not sure how to go about it.