Q1: what does "believe in the system"mean?
He went over every angle, then went over them again from a different direction. He spent time in the passage, then behind the shelves, check-ing the scope of the video camera, testing it. Just a backup now, he thought.
If things went wrong, he had a secondary backup.
“You’re questioning yourself,” Abra said as she checked the lines and fit of the dress she’d put on over a black tank and yoga shorts.
“I used to believe in the system, absolutely. I was part of the system. Now I’m going around it.”
“No, you’re working through it, just in a different way. And even that’s a testament, Eli, when the system failed you. You have a right to defend your home, and a right to do whatever you can to clear your name.”
She added earrings not only to complete the look, but because they boosted her confidence. “You even have a right to enjoy it.”
“You think so?”
“Yes, I do.”Q2:what does "go around it"mean?

Question

Q1: what does "believe in the system"mean?
He went over every angle, then went over them again from a different direction. He spent time in the passage, then behind the shelves, check-ing the scope of the video camera, testing it. Just a backup now, he thought. 
If things went wrong, he had a secondary backup. 
“You’re questioning yourself,” Abra said as she checked the lines and fit of the dress she’d put on over a black tank and yoga shorts. 
“I used to believe in the system, absolutely. I was part of the system. Now I’m going around it.” 
“No, you’re working through it, just in a different way. And even that’s a testament, Eli, when the system failed you. You have a right to defend your home, and a right to do whatever you can to clear your name.” 
She added earrings not only to complete the look, but because they boosted her confidence. “You even have a right to enjoy it.” 
“You think so?” 
“Yes, I do.”Q2:what does "go around it"mean?

Answer

Believe in the system means that you wholeheartedly believe in it and are happy to be in this social setting.  
Go around it means that you may or may not believe in it anymore, and are finding ways to get the to same goal, but in a different way,

Answer

There's not enough context in this passage to know exactly which 'system' he is talking about. I guess that it refers to something that happened previously in the story. 'The system' usually means something like the government system, or the justice system. Where did he work previously? If he was a policeman, he would have been part of the justice system. And if he was part of that system, he would have trusted that it worked properly. He would have 'believed in the system', and believed that everything worked the way it should. It sounds like he has been let down by that system - but you should know that if you have read the previous parts of the story.
'go around it' means that he is not following the normal rules of the system. He is probably not breaking the rules or processes, but more bypassing them, or avoiding them.

Answer

Which book is it?  It sounds like a popular crime novel.