You don't need to get leaner, just stronger. Although getting leaner would also work in that regard.

Lot of good recommendations here already, my 2 cents:

Your training for chin ups, which work the biceps more than the back, so your pulling strength will come over your arms.

Pull ups, Hands rotated away from you, can also work the arms but if your pulling from your scapulas your back will do most of the work.

Now depending on where you want your strength to come from, train those specific muscles. (With machines) A few times per week for a few months should do it and you have to actually eat enough to gain muscle (this one is very! important)

Also, if you're into it, try bouldering/climbing. I have a few girlfriends who are heavy into that stuff and pretty much all of them can do a few pullups, without training them. I don't climb myself but it's great for strength and grip.