Alright, here's the good news. Even the 5th rep on that set wasn't very hard. It moved fast, and you had no trouble with it. You have plenty of strength to spare here, so to speak.
Your observation about your back is correct. But that's simply because you're not taking the time or effort to squeeze up hard before every rep. You do a little half hearted squeeze up. Take an extra half second, squeeze up hard, and this will get better.
Also be sure to finish your lockout completely, and don't bend your arms at the top.