Do English-speaking Jews read the Bible in English, or is that too Christian for them? Maybe they like to get the gist of the Torah or Tanakh by reading in the vernacular English and then they may read prayerfully in modern Hebrew?
Usually if English-speaking Jews want a translation of the Jewish Bible they'll use a Jewish translation.
As for exposure to the other part of "the Bible", i.e. the New Testament, my guess is some Jews who believe in extra exposure will have read bits of it in a survey course or the like.
There's a huge range of degrees to which Jews are exposed to general society, and this means that while most probably haven't read a King James Bible cover-to-cover, often they've come across both "Old Testament" (i.e. Jewish bible) and "New Testament" (Christian) phrases that were introduced by the King James and made it into popular English (or at least college-level literary English).
I was once speaking with a math professor who fully practiced Judaism; she said something about honors "for a student who can walk on water!" -- then she paused and laughed about the turn of phrase, especially when speaking with me. Similarly those who've been through first aid or CPR classes have likely heard of "good Samaritan" laws that legally protect people trying to help. Some people probably know where that phrase comes from, and a lot probably shrug it off without even thinking about it. (Personally I'm fine discussing the good Samaritan laws as such, but when the Talmud talks about conflicts between traditional Jews and another sect trying to mess up their calendar, I like to smile and say "the good-for-nothing Samaritans.")
So I'd say that Jews who have literary exposure will occasionally use such phrases, occasionally with a wink. As long as it's clear we're not endorsing the New Testament as divine. Just as we might use a phrase that originated in Greek mythology or Shakespeare. (The Talmud says Rabbi Meir taught lessons from the book of Proverbs using all sorts of fables involving foxes; that sounds an awful lot like someone else.)
Those who believe in very sharp boundaries against reading anything "from the outside" will probably try to avoid such phrases, though they may unwittingly wind up with some of them anyhow.
Do English-speaking Jews read the Bible in English, or is that too Christian for them?
First, English is not a Christian language. If anything, it's a pagan language, but really it's just a language. There are many translations of the Bible into English. Most are explicitly Christian. A handful are scholarly works (e.g., Everett Fox's translation). Some are explicitly Jewish. The Jewish ones tend to use older Jewish translations and traditional Jewish commentators, like Rashi, Ramban, and Metzudot, to understand difficult verses or words.
If a Jew were to read a translation of the Tanakh into English, they'd likely turn to one of the Jewish translations. Some examples are "Old" JPS, New JPS, Hertz, Stone, and Aryeh Kaplan's translation. Wikipedia has a non-exhaustive list of Jewish translations. As you can see from the number of translations into English, many Jews do sometimes read the Bible in translation.
However, there is serious contention in some parts of the Jewish world if this is a valid practice. Translation of the Pentateuch has been an especially contentious issue in Judaism throughout its history because we are loathe to give up our relationship with the exact words of the Torah. The Septuagint was allegedly written by seventy knowledgeable rabbis under duress. There are several early translations into Aramaic (each called a Targum - Aramaic for translation) which are considered authoritative by modern Jews, and some of our interpretations of difficult verses come from these targums. But as I said, we're loathe to give up our relationship with the original Hebrew, so there are folks who oppose any translation aside from the targums.
Maybe they like to get the gist of the Torah or Tanakh by reading in the vernacular English and then they may read prayerfully in modern Hebrew?
In the vast majority of synagogues of any Jewish denomination, we chant the week's readings aloud in Hebrew. We don't use modern Hebrew, as that would still be translation (and all translations are interpretations), and because that is not necessarily a language we understand. We read the Bible ritually in the language it was written in - ancient Hebrew.
Some have a practice of reading the weekly Torah portion twice in the original and once in Aramaic translation. This is, in fact, a recommended practice, even (especially?) among those who don't believe in vernacular translations.
In practice, those who read English translations of the Torah might read the English to themselves as the Hebrew is chanted in synagogue. They might use the English first when trying to understand the text or might only use English when trying to skim it. They might use the Hebrew first and fall back to English when they don't recognize a word (though this can be problematic). They might read the Hebrew text but the English commentaries. Individual practices, as usual, vary widely based on ideology, language ability, experience, and preference.
So to answer your question, some Jews read English translations of the Bible. Some don't. Most congregations read it ritually in ancient Hebrew. But as a general rule, Jews do not read Christian translations of the Bible for non-scholarly purposes.