I have personally had difficulties with relatives in the past, and I have found the best way is to try and not repeat past mistakes, which you seem to be working on already, and also to "clear the air" about an old situation you may think to still be affecting your relationship. If possible, try and make plans with the person for coffee or lunch. I find that a shorter time is better in case things go poorly, you're not trapped in a long time commitment. Then, try starting with something like:
Family is important to me. I know that in the past I have had some issues expressing that to people, especially you, and I want you to know I'm really working on changing that.
It seems that this is true for you based on your statements, and then allowing the person to respond will be the next step.