Good question,
Php 2:13 For it is God which worketh in you both to will and to do of his good pleasure.
Or, does he motivate you to do, and if so, by means of what?
The same Faith that gave us reconciliation is the same method by which we get His life.
Eph 2:8 For by grace are ye saved through faith; and that not of yourselves: it is the gift of God:
The way to have the life of Jesus is to ask for it. Remember its the Spirit that gives us Jesus life.
Luk 11:13 If ye then, being evil, know how to give good gifts unto your children: how much more shall your heavenly Father give the Holy Spirit to them that ask him?
If we ask believing then it will be ours.
Hope that answers your question.
Both the work of reconciliation and overcoming sin are 100% the work of Christ for us and in us.
The only thing I would add to round out what you have said there is as follows:
Galatians 5:6 "For in Jesus Christ neither circumcision availeth any thing, nor uncircumcision; but faith which worketh by love."
And for this next verse I will use the Bible in Basic English as then it can be easier seen that the word "contraineth" as used in the KJV and more than a few other translations does not mean to hold us still in our tracks but actually means "compels" us but in a way that contains or constrains on the right path:
2 Corinthians 5:14 "For it is the love of Christ which is moving us; because we are of the opinion that if one was put to death for all, then all have undergone death.."
See how that fits with this: 1 John 4:19 "We love him, because he first loved us."
So, does he motivate you to do, and if so, by means of what? Yes and by love.
We must be moved by something to be fruitful and that something is God's love exhibited to us in Christ.
That is how: Php 2:13 " For it is God which worketh in you both to will and to do of his good pleasure", as you said.