The KJV may give the general sense of a translation in many cases. However, it has a definite Christian bend and does not always follow the Jewish traditions in translation. I will illustrate this phenomenon with a few examples.
Isaiah 7:14 (see here):
MT:
לָ֠כֵן יִתֵּ֨ן אֲדֹנָ֥י ה֛וּא לָכֶ֖ם א֑וֹת הִנֵּ֣ה הָעַלְמָ֗ה הָרָה֙ וְיֹלֶ֣דֶת בֵּ֔ן וְקָרָ֥את שְׁמ֖וֹ עִמָּ֥נוּ אֵֽל׃
KJV:
Therefore the Lord himself shall give you a sign; Behold, a virgin shall conceive, and bear a son, and shall call his name Immanuel.
NJPS:
Assuredly, my Lord will give you a sign of His own accord! Look, the young woman is with child and about to give birth to a son. Let her name him Immanuel.
KJV translates עלמה as "virgin"—in "anticipation" of the Christian belief that Mary was a virgin when she begot Jesus—while NJPS translates this term in the usual Jewish way, i.e. "young woman" (virgin = בתולה).
Isaiah 40:3:
MT:
ק֣וֹל קוֹרֵ֔א בַּמִּדְבָּ֕ר פַּנּ֖וּ דֶּ֣רֶךְ יְהוָ֑ה יַשְּׁרוּ֙ בָּעֲרָבָ֔ה מְסִלָּ֖ה לֵאלֹהֵֽינוּ׃
KJV:
The voice of him that crieth in the wilderness, Prepare ye the way of the Lord, make straight in the desert a highway for our God.
NJPS:
A voice rings out: “Clear in the desert A road for the Lord! Level in the wilderness A highway for our God!
The KJV translation is based on the interpretations in the LXX and the Christian canon (MattewMatthew 3:3; Mark 1:3; John 1:23) where John the Baptist is described as "the voice of one crying in the wilderness" (φωνὴ βοῶντος ἐν τῇ ἐρήμῳ). However, the NJPS translation agrees with the syntax of the te`amim.
KJV Ps. 22:16 = MT Ps. 22:17 (see here):
MT:
כִּ֥י סְבָב֗וּנִי כְּלָ֫בִ֥ים עֲדַ֣ת מְ֭רֵעִים הִקִּיפ֑וּנִי כָּ֝אֲרִ֗י יָדַ֥י וְרַגְלָֽי׃
KJV:
For dogs have compassed me: the assembly of the wicked have inclosed me: they pierced my hands and my feet.
NJPS:
Dogs surround me; a pack of evil ones closes in on me, like lions [they maul] my hands and feet.
Here the KJV follows the LXX reading, and not that of the MT. It is obvious that the this translation was used to seem like Psalms is "anticipating" the crucifixion of Jesus.
The list goes on. The KJV will also occasionally diverge from the Masoretic verse divisions (e.g. the example above, MT Num 25:19 = KJV Num 26:1, etc.).
At any rate, I find NJPS is a good Jewish translation, in precise and modern language. The 1917 JPS version, which is somewhat outdated, is not bad and is easily found online. Artscroll and Koren are popular Jewish translations too.