The Matou family has a long history involved with the grail war, but hasn't had a magic user strong enough to participate in the last two generations. This also includes Shinji. His "grandfather" considers him and his father worthless. He is expected to perform as a magus despite lacking a magic circuit.
Shiro and Shinji were previously good friends, but Shiro noticed a change in Shinji over the years. This is due to the Matou family's grail war training causing Shinji frustration in his inadequacy. Shinji grew distant, and hot tempered. Things finally reached the boiling point when Shinji hurts his sister Sakura. The two had an intense argument and now rarely speak to each other. On friendly terms anyway.
In the grail war Shinji offers alliances to both Rin and Shiro, however neither trust him. This alienates him even more. With his bad temper, poor strategy and lack of courage; Shinji is a relatively weak opponent but causes problems when opportunity presents itself. He especially holds a grudge against Rin. Probably because she jilted him, but also due to jealousy over her talent in magic.
What's the fun of making a guy beg?
I brought you here to fight.