The actual verse
Philippians 4:13 in the BSB:
I can do all things through Him who gives me strength.
In the KJV (1769):
I can do all things through Christ which strengtheneth me.
The two renderings differ on whether the pronoun “him” or the proper noun “Christ” appears in the verse. This is a textual question — the manuscripts of the Greek New Testament disagree at this point — not a question of theology.
The surrounding passage
The verse appears at the close of Paul’s letter to the church at Philippi. The full immediate context (Philippians 4:10–13, BSB):
10 I rejoiced greatly in the Lord that at last you have revived your concern for me. You were indeed concerned, but you had no opportunity to show it.
11 I am not saying this out of need, for I have learned to be content regardless of my circumstances.
12 I know how to live humbly, and I know how to abound. I am accustomed to any and every situation — to being filled and being hungry, to having plenty and having need.
13 I can do all things through Him who gives me strength.
The surrounding verses describe Paul’s experience of fluctuating material circumstances — being filled and hungry, having plenty and having need — and his learned response of contentment in each.
Original language
The Greek of the verse, in modern critical editions (Nestle-Aland 28th edition, United Bible Societies 5th edition):
πάντα ἰσχύω ἐν τῷ ἐνδυναμοῦντί με
Literally: “I am strong [in] all things in the one who strengthens me.”
The Textus Receptus, on which the KJV is based, has an additional phrase ἐν Χριστῷ (“in Christ”):
πάντα ἰσχύω ἐν Χριστῷ τῷ ἐνδυναμοῦντί με
The added words do not appear in the earliest manuscripts (e.g., Papyrus 46, Codex Sinaiticus, Codex Vaticanus, Codex Alexandrinus). Modern translations follow the older manuscripts and render the verse without “Christ” as the explicit noun, while the KJV and NKJV preserve the Textus Receptus reading.
The verb ἰσχύω (ischyō) means “to be strong, to be able, to have power.” The participle ἐνδυναμοῦντί (endynamounti) is from ἐνδυναμόω, “to make strong, to empower.”