Well, I disagree with some methods suggested, and agree with others.
The issue with grip is that you lack forearm strength, wrist strength and finger strength. -Not any one of the three, ALL THREE.
The best fix for eliminating your weak points is to ditch any and all aids such as straps, chalk, etc. -But do so at a much lower weight than you currently lift.
After you do that, and determine the weight you lift safely, then find a bar with a grip your hand only goes 1/2 to 2/3 of the way around. (Generally this is a 2 1/2 or 3" grip.) Again, lower the weight you lift significantly and load up the big bar. Lifting with the big bar ONLY will slowly bring your fingers, wrist, and forearms into line with the rest of your body for functional strength but it will take a while.
The grip is one of the EASIEST things to over work. If you have the pain you claim, back off for a week and only SLOWLY ease into the training again. Remember: you are dealing with an injury point because it is not up to speed with the rest of your mucscles. Once the grip is in shape, and you can again lift what you do today with a big bar, then it would be safe to advance in weight.
You will find a weak grip is prevalent in many weight training circles, and those who work grip often do it wrong. The best way to get good in a single lift is to practice the lift a zillion times. Grip is no different. If you want to grip a bar, don't look to grippers or straps or chalk. Build the grip with the bar. Farmers walks are only 50% effective if done with a small bar. Keep the big bar in use.
You will notice after the grip is strong enough to handle what you presently lift, and if you stay with a big bar, your gains will always come much faster than they did with the small bar. Once you get along where you are happy with strength levels, switch only occasionally back to small bars for practice, but keep using big bar as frequently as possible for gains.
Aside from all of this, in order to maximize the grip, I would eliminate ANY barbell use. Going with dumbbells alone will cause the fingers and wrist to see more action than barbells. With a wide grip on dumbbells, you will soon see why those are the manliest lifting tools ever made.