Covenant theology in confessional Reformational theology is its hermeneutical key. Karl Barth, a Reformed theologian, doesn’t stray from this key, but as is typical with Barth he reformulates Covenant (or Federal) theology such that Jesus Christ becomes the key, the regulative ground and condition of the covenant itself. Indeed, rather than operating with 
two aspects of the covenant—i.e., the covenant of works, covenant of grace—as classical Covenantal theology does, Barth retextualizes this framework by reducing the two covenants into one; viz. the covenant of grace. For Barth, the covenant of grace is the supralapsarian (before/above creation)/fall) basis, the inner reality of creation’s concrete expression, as God in pre-temporal reality pre-destined Himself to be for and with us in His election of our fallen humanity for Himself; and thus, as corollary, in this election we also, as humans, in Christ’s elect humanity, come to experience the exalted status of what it means to be human before God in the resurrection of Jesus Christ. This is God’s covenant of grace, His free election to be God with us, rather than against us; it is this election that has consequences not just for us, but as we have been called to be priests of creation from Christ’s priesthood, it has consequences for all of creation’s redemption and re-creation (cf. Rom. 8:18ff). But it is this framework that Barth reifies, this covenantal framework, and it becomes one conditioned not by a naked heilsgeschichte (‘salvation-history’) shaped by the abstract decretum abosolutum (absolute decree of predestination), but by the concrete reality of God’s inner-triune life that has freely been given for the world in Christ.
It is the aforementioned framing that Barth takes up in discussion on the Sermon on the Mount vis-à-vis the covenant of grace. There is a biblical specificity to the way Barth attempts to explicate theological reality, and the following is an example of that. It isn’t polemical, but a constructive way to think about the implications of the Incarnation and how that impacts the way we think all things biblical. If nothing else this ought to illustrate how Christological Interpretation of Scripture (CIS) looks, particularly as that gets fleshed out in a foundational way in regard to the way that Barth sees the promise/fulfillment motif in Jesus Christ as the grundaxiom of biblical and Christian reality. He writes:
The Sermon on the Mount, too, is primarily and decisively a notification, a proclamation, a description and a programme. Its imperatives, too, have primarily and decisively the character of indicating a position and laying a foundation. The position indicated and the foundation laid are the kingdom, Jesus, the new man. And these are not three things, but one and the same. The kingdom is the new man in Jesus. Jesus Himself is the kingdom of the new humanity. The new Adam is Jesus the Bringer and Herald of the kingdom. Proclaiming this threefold unity, the Sermon on the Mount proclaims the consummation of the covenant of grace, and therefore the telos of the Law and the Ten Commandments. It proclaims the position which in the Ten Commandments was determined and promised to Israel, but only determined and promised and not given. The Sermon on the Mount, like the New Testament as a whole, defines and describes it as something now given. If the Ten Commandments state where man may and should stand before and with God, the Sermon on the Mount declares that he has been really placed there by God’s own deed. If the Ten Commandments are a preface, the Sermon on the Mount is in a sense a postscript. The history of the covenant of grace has reached its goal and end. It does not continue in the history of the Church at whose beginning there stands the declarations of the Sermon on the Mount. For time does not continue after the death and resurrection of Jesus Christ. It can only move away from this its centre—moving to its already appointed end. For its end is determined by the fact that it has been given this centre. The Church is the community of God in this time which moves away from this centre to its appointed and already invisible end. The covenant of grace as such has no further history in this time. The only question now is whether the Church will live or not live in the fulness of life already granted to it, in recognition or non-recognition, gratitude or ingratitude, in face of what God has finally and once for all accomplished for man, in the freedom which God has decisively accorded to man, or the bondage from which he has been finally and conclusively released, and which has now become a complete anachronism. The Sermon on the Mount, as a postscript, as a document of the completed covenant of grace and its concluded history, defines and describes the freedom which is given to the people of God in its new form as the Church, and which is to be proclaimed by the Church to the whole world.[1]
For Barth (and I’d suggest for the Apostle Paul) Christ is the new creation (cf. II Cor. 5:17). This isn’t an abstract notion we have been called simply to affirm, but a concrete reality to be understood and lived from. This is what we see Barth explicating in his exegesis of the Sermon on the Mount. Here Barth is engaging in a canonical exegesis, he is using the Pauline motif of new creation, within a Christological frame, to interpret the dominical teaching of Jesus deposited in the Sermon on the Mount. When Barth refers to the Sermon on the Mount as a postscript, he, in context is referring to the promise of the protoevangelium (Gen. 3:15). He is thinking in terms of his teaching on new time, in corollary with Christ in His humanity as the new creation. It is God’s choice, prior to the foundation of the world, to be for and with us in Christ, as the inner reality of all reality, that is finally consummated for Barth in the death, burial, resurrection and ascension of Jesus Christ. As such insofar that creation has Christ as its telos (purpose), as its res, all time has been brought to its end; that is in the sense that it has finally come into its glorified intention. For Barth, eternity and time are not in competition, but in fact the former, in God’s election in Christ, is the latter’s basis. It is this basis, the new time founded in the new creation of Christ’s resurrection and ascension that God’s community of people, the Church, finds their reality. As such we no longer are scrapping out an existence under the old strictures, only looking forward to the kingdom of God coming, instead we are now living in the reality that the kingdom has indeed already come, and yet will finally come in consummate form. But it is this consummate form that Barth wants us to press into and live from as if it is already the reality; this is the new time of the kingdom, and the center of all reality, as such. The Sermon on the Mount is the postscript insofar that its reality, as a summary of all the Law and Prophets as now fulfilled in Jesus Christ, is indeed indicating the finality of Kingdom Come!
[1] Karl Barth, Church Dogmatics II/2 §36-39 [688] The Doctrine of God: Study Edition (London: T&T Clark, 2009), 175.