Pursuing holiness means pursue it in Christ.

"Follow peace with all men, and holiness, without which no man shall see the Lord: 15Looking diligently lest any man fail of the grace of God; lest any root of bitterness springing up trouble you, and thereby many be defiled; 16Lest there be any fornicator, or profane person, as Esau, who for one morsel of meat sold his birthright. 17For ye know how that afterward, when he would have inherited the blessing, he was rejected: for he found no place of repentance, though he sought it carefully with tears." - Hebrews 12



"The second thing to be pursued is 'holiness'. The Greek word uses here is sometimes translated 'sanctification', and many consider it refers to 'progressive sanctification'. Thus Kistemaker comments, 'The word in the original Greek refers to the sanctifying process that occurs in the life of the believer . . . the believer . . . becomes more and more like Christ.'
However, we need to be cautious here. Of the five cases where this word is translated 'sanctification' in the New Testament (AV), most refer unambiguously to 'positional sanctification' -- the believer's standing as one 'set apart' in Christ (for example, 1 Cor. 1:30; 2 Thess. 2:13; 1 Peter 1:2). Furthermore, if 'progressive sanctification' is intended here, salvation itself (seeing the Lord) becomes dependent on such sanctification. What, then, of those who repent on their deathbeds and have no opportunity to 'progress' in sanctification? And how far beyond justification by faith must we progress before we are qualified to 'see the Lord'? A theological quagmire confronts us!
The context, therefore, seems to rule out 'progressive sanctification' as an explanation of this exhortation - as other commentators have recognized. Lane declares, 'In Hebrews human endeavor is never the subject of sanctification. Christ alone is the one who consecrates others to God through his sacrificial death. Holiness is . . . the objective gift of Christ achieved through his sacrificial death on the cross (10:29; 13:12).'
Where does this leave us? It means that the injunctions to 'pursue peace' and to 'pursue holiness' are emphasizing the same need -- to strive to lay hold upon the gift of God, the pure promise of grace in Jesus Christ. He alone is our peace, our sanctification, our holiness, and we are 'complete in him' (Col. 2:10). To seek these things outside of him -- in ceremonies, in shadows, in law-keeping, or anywhere under the old covenant -- is to turn aside from the race that is set before us, to depart from the Highway of Holiness." - Edgar Andrews, A Glorious High Throne