How does Priestley show the realationships between the characters?

This is an awfully tall order for a short answer forum. I suggest you begin by reading through gradesaver's study guide, and if you still have questions afterwards come back for something more specific.

http://www.gradesaver.com/an-inspector-calls/

Eric and his parents- They have a complicated relationship, Eric drinks way too much and his mother can admit to it and admit to him having sexual relationships between him and Eva. When the inspector talks to Eric, He is drinking far to much and he Mr Birling doesn't want Eric to say something he shouldn't because of the drink, He also can't have much trust for his son once he found out he stole some money from his own factory.

Shelia and Gerald- Shelia can't trust Gerald that much anymore after she found out that he had Eva as his mistress for 6 months. However, she is willing to hear his involvement with Eva. Shelia and Gerald was going to get married and their engagement was ready however he broke the trust therefore broke down the marriage in some sorts.

Shelia and her parents- I'm not awfully sure about the relationship between her and her parents however her mother still treats Shelia and Eric for that matter as they are still children and mothers them. (You'll have to find out more about that i don't know)

Gerald and Mr Birling- Mr Birling said Gerald is like him a smart buisness-man who is good in the line of work he does. He hs great respect for Gerald and his family, maybe that is because they are in upper class and the Birlings are in middle class or he is generally the same as Mr Birling (personality wise) Gerald does agree at the end with Mr Birling that he was right to fire Eva and they both agree on that.