I remember that the translators of the King James Bible knew Hebrew, is it so outlandish to think that some Christians would have read/studied it as well?

You need a lot more background than that to read the Talmud. First, a great deal of it is in Aramaic, which is much less well-known than Hebrew, at least among western Christians. And Hebrew wasn't exactly common knowledge among Christians. And biblical Hebrew and Rabbinic Hebrew are somewhat different. Second, even if you understand the words, it has a great deal of technical vocabulary that's rather opaque. Terms that structure passages, legal terms, etc have very non-obvious meanings, and require a lot of in-depth knowledge. Not to mention a wealth of acronyms that can befuddle most anyone. Third, and perhaps more importantly, it requires a lot of context to understand. The Talmud is essentially an in-depth discussion of Jewish practice. Unless you already have a solid background of that, it's going to be difficult to understand. I can't say none did (and I imagined it happened occasionally), but it would've been rather rare.

That said, some of the knowledge from it did filter through to Christian theologians. One commentator, the Radak, was translated to Latin and read by Christians on occasion. His commentary on the Torah makes heavy use of exegesis and explanation from the Talmud. Of course, modern academics who are Christian do sometimes make use of the Talmud, but I don't think that's what you're asking about.

Do you mean the Torah, Tanakh, Mishnah, or Talmud? The Torah is the books of the law. The Tanakh is equivalent to the Christian Old Testament and includes the Torah. The Mishnah is a collection of oral tradition and law fixed in 200AD, and the Talmud is the Mishnah plus the Gemara, a set of commentaries on the Mishnah. The Talmud is large - Wikipedia estimates 6300 pages.

Christians have always studied the Tanakh in translation, and theological colleges often teach Hebrew (as well as koine Greek) to allow some study of the original text.

The Mishnah has historically not been much known to Christians, but does provide some important views for interpretation. I'm not sure when the Mishnah became studied by Christians, but currently it does have a certain amount of influence, mainly to guide interpretation of the New Testament. The rest of the Talmud is less used, and much more specific to matters which are specific to Jewish faith and practice.

Christian use of the Talmud is very largely indirect, so that a Christian is much more likely to read of a Jewish interpretation of Messianic prophecy through reading a non-Hebrew commentary than to read a translation directly, even though English translations of at least the Mishnah have been available for quite a long time - 1933 for the Danby translation, for instance.

To give a timely example of information from the Talmud which can influence Christian interpretation: we are approaching Pentecost. The Christian understanding of this is that at this time the Holy Spirit came on the disciples in a form like tongues of flame, giving them the ability to speak in the languages of all nations. The Talmud gives a context to this: at Pentecost (i.e. fifty days after the original Passover in Egypt), Moses received the law. At the same time, seventy tongues of flame went out to all the nations, speaking the law in their own tongues. This is just an illustration of where some familiarity with the Talmud is occasionally used in a Christian setting.

EDIT - for clarification: the Talmud is not treated as inspired in Christian tradition, and there are some other ancient non-canonical resources which are used in the same way. Examples would be the Dead Sea Scrolls, and works such as 1 Enoch and the Gospel of Thomas. Use of these documents is more common in the Protestant tradition, which depends more on exegesis and less on church Tradition.