How does this outlook relate to his experience in the woods and to his overall view of how life should be lived?

In "Civil Disobedience," Thoreau claims that we have a moral obligation to refuse to obey unjust laws. Further, we have a responsibility to not contribute in any way to immoral institutions, such as slavery. For example, Thoreau refused to pay his poll tax, a tax levied on all citizens regardless of income, because he could not suffer his money being used to support a government that embraces the institution of slavery or an unjust war with Mexico. This act of civil disobedience got him sent to jail for one night (at which time someone else paid the tax for him). He urges all of us to evaluate the ways in which we might be inadvertently supporting unjust practices and to remove our support in whatever way possible: verbally, socially, financially, and so on.
In his other writings, Thoreau urges a similar kind of purposefulness. He wants us to really think through everything we do to make sure that we really want to be doing it and aren't simply going through the motions of life. We must "live deliberately." He wants us to understand the many and varied implications of the things we do, both for ourselves and for others. When we purchase a large home, for instance, we've saddled ourselves with the obligation to heat it, furnish it, keep it clean, and so on. When we pay our taxes, we might inadvertently be supporting unjust practices. It all comes down to being deliberate and purposeful and thoughtful; we cannot passively act and do. We must consider carefully how to live simply and honestly and ethically.