Kill leader is a bit more complicated than it may seem. Although it is still very simple. To put it straight forward it seems the first to get 3 kills is awarded kill leader. Then after that You need to get more kills then the kill leader.
If you kill the kill leader and have 3 or more kills, you should become the kill leader regardless if someone has more kills. If you have more kills in that situation, you will have to get another kill to register you as the kill leader.
If you wanna know how many kills the kill leader is currently at, there are screens all around that cycle through the champion and kill leader and will show their kills.