The concept of great expectations has to do with having a big hope that someone wants to have fulfilled. In this chapter, the local police are getting out and after some escaped convicts. Their cause is great. With such a great expanse of land and such a set of tricksters on the loose, it would be difficult to locate them. The police also have great hope in the reality of finding these men. They have an attitude of expectation for which they do not want to accept anything less than their goal.
If you are asking what the chapter has to do with the entire storyline, this chapter certainly reveals a particular convict who works to get Pip out of terrible circumstances to bring him to the highest possible point. Had the convict not confessed to stealing many items from the Gargery's home, Pip would have suffered Mrs. Joe's wrath.
Pip too had hope that his circumstance would work out in his favor. The benefit of this single episode in chapter 5 is that it does work. That doesn't necessarily mean it will continue to work for him.