Okay explain to me how the system is developed/started, and how it is improved
I can't believe you're seriously asking this question on a forum, it's like asking someone how to engineer a submarine on a public forum, the answer being many volumes of mathematics and physics etc... The question can be directed to qualified men of law and expert jurists.
When `Umar created the Diwan and established it as a system to spend on certain deserving categories of people in the Islamic state, this system was not available in Abu Bakr's reign. Therefore, this is how that form of government improved and evolved. Keep in mind that the method of producing and organizing the Diwan was never stated in any Islamic divine text, it was human effort, Islam did not dictate a specific system for this.
how the people choose the leader
Depends on the system in place. In some countries leadership is inherited, in others a council of experts chooses, in others all people vote and choose etc... Obviously the contitution needs to specify what are the qualifications for a person to run and be selected in the first place. An Islamic state can place the rule that the Imam must always be Muslim.
how do you know a person or “leader” is mujtahid and his level of ijtihad?
You mean a jurist or scholar? Scholars are usually recognized by all other scholars and laypeople in their areas. I don't think society with its learned men and educated scholars will have any issues figuring out if a man is a scholar. In the US, there are presidential debates for instance, these are televised and each nominee is judged by the expert panel and the people.