How can I overcome sin in my Christian life?

Overcoming sin is a big job, but it can be done. I can’t answer for all denominations, only the one that I follow. I belong to the Church of Jesus Christ of Latter-day Saints, so I’ll stick to what we do to overcome sin.
The first thing we do is recognize that we’ve committed a sin.

Then, we look at whether we feel regret for what we’ve done or if we only feel shame. If we only feel shame, that’s not good; that’s Satan attempting to keep us from taking the next step, which is to start making restitution. We apologize, go to see the bishop, or both, depending on the sin. Then, we make use of the atonement and let the sin be washed away.