Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed. (1 Pet 2:24)
This is a common "proof-text" to support penal substitution and, as a result, the forensic nature of justification; according to many Reformed apologists, this verse shows that sins (at least of those of the "elect") were imputed to Christ's body on the cross. However, this is gross eisegesis.
The verb translated as "bare" is αναφερω, which is a sacrificial term. This verb is used ten times in the Greek New Testament in contexts of "lifting up" or "offering sacrifices" (e.g., Matt 17:1; Mark 9:2; Luke 24:51; Heb 7:27; 9:28; 13:15; 1 Pet 2:5). Note how the verb is defined in BGAD (highlight added for emphasis):
3. to offer as a sacrifice, offer up, specif. a cultic t.t. (SIG 56, 68; Lev 17:5; 1 Esdr 5:49; Is 57:6; 2 Macc 1:18; 2:9 al.; ParJer 9:1f; Did., Gen. 219, 15) ἀ. θυσίας ὑπέρ τινος offer sacrifices for someth. Hb 7:27. ἀ. τινὰ ἐπὶ τὸ θυσιαστήριον (Gen 8:20; Lev 14:20; Bar 1:10; 1 Macc 4:53; Just., D. 118, 2 θυσίας) offer up someone on the altar Js 2:21. Of Jesus’ sacrifice: ἑαυτὸν ἀνενέγκας when he offered up himself Hb 7:27. τὰς ἀμαρτίας ἡμῶν αὐτὸς ἀνήνεγκεν ἐν τῷ σώματι αὐτοῦ ἐπὶ τὸ ξύλον he himself brought our sins in his body to the cross 1 Pt 2:24 (cp. Dssm., B 83ff [BS 88f]). Pol 8:1 (Is 53:12).—Fig. (schol. on Apollon. Rhod. 2, 214b χάριν=render thanks to the divinity) ἀ. θυσίαν αἰνέσεως offer up a sacr. of praise Hb 13:15 (cp. 2 Ch 29:31). ἀ. πνευματικὰς θυσίας 1 Pt 2:5. ἀ. προσευχάς offer prayers 2 Cl 2:2. ἀ. δέησιν περί τινος offer up a petition for someth. B 12:7.
Additionally, note the following comment from two biblical scholars writing on the nature of the atonement with respect to related sacrificial terminology used in the gospels:
If the Johannine formula, “who takes away sin,” is understood as a reference to the “Servant of God,” it must be recalled that neither in vv. 4 nor 21 of Is 53 is the verb nāsā accompanied by the phrase, “upon himself,” in spite of the Greek translation in verse 4, tas hamartias hēmōn ferei, “he carries our sins.” When Matthew applies the statement of Is 53:4 to Christ, he wishes to say that he took away our illnesses, not that he took them upon himself. The same meaning is found in the Exultet of the Paschal liturgy: “he is the true Lamb who took away (absulit) the sins of the world,” and in the Epistle to the Hebrews: “Thus Christ offered once for the taking-away of the sins of many, a second time—without sin—will be seen to those awaiting for salvation” [Heb 9:28]. (Stanislas Lyonnet and Léopold Sabourin, Sin, Redemption, and Sacrifice: A Biblical and Patristic Study [Analetcta Biblica Investigationes Scientificae In Res Biblicas 48; Rome: Pontifical Biblical Institute, 1970], 40)
There is no hint at the later doctrine of penal substitution in this passage and others. Note the following from one of the best books refuting penal substitution from both the Bible and the patristics which I highly recommend:
To understand the heavenly intercession of the Son on our behalf as the propitiation of the Father, as Michael does, generates a significant problem of internal coherence for penal substitution. According to penal substitution, the primary purpose and effect of the death of Jesus was to propitiate the wrath of God on account of the sins of humanity. As it is written elsewhere, because Christ is “a priest forever” in heaven, he “always lives to make intercession” and is thus “able for all time to save those who approach God through him” (Heb 7:24-25). Heavenly intercession on our behalf is thus the ongoing vocation of the risen and ascended Christ. So, if the purpose and effect of the Son's intercession is to propitiate the Father's wrath, then the Son is continually doing in heaven at the throne what was to have been fully accomplished on earth at the cross. The cross would thus seem to have been ineffective, or at least incomplete, in accomplishing its primary purpose of saving humanity from divine wrath. Michael's [a Reformed apologist the author is responding to] interpretation of 1 John 2:1-2, although given in defense of penal substitution, effectively undermines it. (Darrin W. Snyder Belousek, Atonement, Justice, and Peace: The message of the Cross and the Mission of the Church [(Grand Rapids, Mich.: Eerdmans, 2012], p. 249 n. 13)
As for the phrase “dead to sins,” see this post which addresses, in part, Total Depravity (the “T” of TULIP).