Question — Can I achieve holiness on my own?

Answer — No. Christ came to die and satisfy God’s wrath because we violated His holiness. In exchange for Christ’s sacrifice, God credits righteousness to all who believe in His Son’s death, shed blood, and resurrection. Though we may strive for righteousness in our Christian lives after salvation, apart from God, our efforts fall short:

But we are all as an unclean thing, and all our righteousnesses are as filthy rags . . . (Isaiah 64:6)

God never gives us a command He does not equip us to obey. While holiness is an impossibility for an unbeliever, with the aid of the Holy Spirit, Christians can develop this trait.

That ye put off concerning the former conversation the old man, which is corrupt according to the deceitful lusts; And be renewed in the spirit of your mind; And that ye put on the new man, which after God is created in righteousness and true holiness. (Ephesians 4:22-24)

If God created our new nature in righteousness and holiness after salvation, He can cultivate it so its fruit is later manifested in our service to Him. We must simply allow Him to work.