You say 5x5 but what weight are you using for your pullups/chinups? Why are you doing a 5x5 trying to get bigger? When Nearly every article, every website, and most posts say that is good for strength more than size?
I gained 40 lbs in six months on a 5x5. Strength and size are pretty much the same thing for most novice and intermediate lifters. If the 5x5 isn't making you bigger, you're probably not doing the exercises correctly or eating enough.
To the OP. Have you gotten stronger? Have you gained weight? If not, you can't expect your back to get bigger. Also, you should probably be doing Pendlay rows instead of chins as your back exercise.
Try adding lbs every time you meet your rep range with either a dumbbell between your feet or a dip/chin belt. I take in 2-3 protein shakes a day, but that's on top of my 4-5 solid meals. So if you aren't eating enough solid food don't be worrying about protein shakes too much.
There's more to back size and strength than pulls and chins. Hopefully, you're doing rows from various angles also. Have you tried weighting yourself when you're doing the pulls and chins? Your body weight may not be enough.
As far as eating goes, I eat eight times a day. And none of it is shakes. Real food.
If you gained 40lbs of muscle in six months I doubt it was the routine. If you did it doing 5 sets of less that 6 pullups/chinups your a genetic freak. For overall body mass going heavy enough 5x5 will help, for trying to hit a specific area like the back and increase its size noticebly as fast as possible I don't see that happening.
I think the OP should probably answer everybody's questions before expecting more help.
One more question is what are you looking for width? Thickness? both?
My point is that as a beginner or intermediate lifter, it doesn't make much sense to target a specific area in hopes of getting it bigger. If you're doing squats, deadlifts, and rows, if you eat enough, and if you add weight to the bar every week, you'll gain muscle everywhere, including your back.