The word "holiness", or "sanctification", is the translation of the Greek hagiasmos, used in the Bible under two aspects:
A doubt sometimes arises as to which aspect is applicable in the exhortation given in Hebrews, for if we follow the argument in the preceding verses, where the writer talks about God's discipline, who chastens us "for our profit, that we may be partakers of His holiness", we get the impression that it has to do with our own effort. But, as without this holiness no one will see God, he must be referring to the first aspect.
The answer is that both aspects are included here. There is only one holiness, and not, so to say, a type which means purification and dedication, and another type limited to the day-to-day effort of the believer in his battle against the flesh, the world and the devil. No one can, by himself, attain holiness: he cannot even separate himself unto God on his own, let alone separate himself from his own lusts, the world and the devil.
Holiness is, in essence, a relationship with God. The only way to enter this relationship is by faith in Jesus Christ, in the Spirit of God (Acts 26:18; 1 Corinthians 6:11), and it only became possible by means of His death in atonement for sin (Ephesians 5:25, 26; Colossians 1:22; Hebrews 10:10, 29; 13:12). This is why the verse in question says that, without holiness, "no one will see the Lord." Whoever does not have this relationship with God cannot enter in His holy presence.
Having thus entered in this relationship, the believer is as if "adopted as son" by God. God wants to make the believer's holiness perfect (1 Thessalonians 4:3), separating him from evil ways and things because it was for this reason that the Gospel was brought to him (1 Thessalonians 4:7), and it is from God that he is going to learn (1 Thessalonians 4:4) in the measure in which God teaches him by His Word (John 17:17, 19). Like a father chastens his son in order to guide him with wisdom until he reaches maturity, so does God chasten us for our benefit in order that we may be partakers of His holiness.
He teaches us by His Word, in the practice of which we gain experience, and in the disobedience of which we suffer punishment. Within our local assembly we learn and educate each other in the way to holiness.
Our faith is tested for our strengthening: the trials are bearable, as we are not tested beyond our endurance; they are hard and can bring us suffering, but there will be joy with the victories. We also have, for our encouragement, a reward promised to each one who overcomes.
Although we have been given the power of the Holy Spirit to attain victory, we are subject to becoming tired, discouraged and even to fall. This passage in Hebrews urges us to persevere in the midst of the discipline of the Lord when we are reproved by Him. A chastening does not mean abandonment, on the contrary, it is an integral part of our learning process and proves the great loving interest of the Lord in His high purpose of making us participants of His holiness.
We are exhorted to strengthen the hands which hang down and the feeble knees of those who are weakening; to make straight paths for your feet, so that what is lame may not be dislocated, but rather be healed: let us consider those who have difficulty in correctly learning the will of God, and teach them clearly so that they may know how to conduct themselves, and be a good example unto them; pursue peace with all people (if it is possible, as much as depends on you - Romans 12:18), and holiness, without which no one will see the Lord, for which we depend entirely on our God and Saviour, Jesus Christ.
As every believer is sanctified in Christ Jesus (1 Corinthians 1:2), every believer is called a saint in the New Testament. Therefore holiness, or sanctification, is not something which is attained upon the merits of a person, but it is a state in which God freely places sinners, and in which they begin their Christian career (Colossians 3:12, Hebrews 3:1).