Quite often, Christians will get accused of relying on themselves instead of Christ[/quote
Relying on ourselves does not involve relying on anyone else, so it would be contradictory for someone to think that we can rely on ourselves by relying on God's instructions. God is trustworthy, therefore His instructions are also trustworthy (Psalms 19:7), so the way to rely on God is by obediently relying on His instructions, and it would contradictory for someone to think that we should rely on God, but not on His instructions. God did not give His law as instructions for how to save ourselves apart from Him, rather God graciously teaching us to obey His law is the way that He is giving us His gift of salvation (Titus 2:11-13).
who believe in salvation by faith alone
Luther said that an idle faith is not a justifying faith, so His doctrine of faith alone should not be understood in a way that is contrary to His position. A couple can be alone apart from the company of others while not being alone apart from the company of each other, so someone can be alone and not alone at the same time in different senses. The position that we are declared righteous by faith alone is apart from being required to have first done enough righteous works in order to earn it as the result, but is not alone apart from becoming a doer of righteous works. To say that God is righteous means that He is a doer of righteous works, so while the only way for someone to become righteous is through faith apart from works, it would be contradictory for someone to become righteous apart from becoming a doer of righteous works. This is why the same faith by which we are declared righteous does not abolish our need to be a doer of righteous works in obedience to God's law, but rather our faith upholds it.