As a Japanese beginner I am learning simple phrases like "Watashi wa Chris desu" to mean "I am Chris" but literally it would be more like "I Chris am" right? Another example would be "Genki desku ka?" meaning "How are you?" but literally would be more like "Good are/am?" right?
My point is I feel like it would be much better not to focus on translating Japanese sentences into the English equivalent because it forces you to THINK in English and not in Japanese. I feel like I should be training to think like a Japanese person and so should think more in literal word for word translations (like "I Chris am" and "Good are/am?" etc.) and not in English translations of the meaning.
I think it can be enlightening to think about the literal translation, because it can give you insight into how the language works, but I'm not sure you should really think about the literal word-for-word translation instead of the natural English translation. Your goal is to use Japanese naturally, and I do rather suspect that making a habit of thinking about Japanese in that very unnatural way might be a hindrance to that.
As your Japanese skill improves, you will naturally rely less on the English translation anyway, and be more able to think in Japanese to begin with. That'll happen without forcing it, so forcing it seems like a bit of a risk. In the meantime, try thinking in English and then translating it into Japanese. You will make mistakes, but then that's the point of learning, isn't it? The more mistakes you make, the better you'll understand Japanese, and the more you'll be able to think in Japanese without first thinking in English.
additionally to what 小狼-san said, I would like to add, that the bit about the literal translation is a bit more tricky. As you get deeper into the japanese language, you get more accustomed to the concept of omitted subjects and objects. So, both of your example sentences simply have omitted subjects.
Like "watashi wa chrisu desu" would USUALLY be "watashi wa watashi ga chrisu desu". Everything before wa siply is the TOPIC of the sentence. It's not the subject. The subject simply is omitted, since it's the same as the topic. So, the LITERAL translation of "watashi wa chris desu" would be: "As for me, I am Chris."
The same is true with "genki desu ka", where anata ga simply is omitted. So the sentence would usually be "anata ga genki desu ka.", or, even more complete: "anata wa anata ga denki desu ka", so the literal translation would either be "Are you well?" or "As for you, are you well?"
Japanese just doesn't work that way, because the subjects (and objects) often are omitted. But just that they're omitted doesn't mean that they're not THERE!
So, one reason why I think that you SHOULD translate a sentence always to it's correct english (or whatever your language is) is so that you learn the true meaning of the sentence and get more aware of what exactly got omitted.
So, a dialogue like this:
A: "o-genki desu ka." B: "genki desu."
Is just the japanese-omitting.version of:
A: "anata wa anata ga o-genki desu ka." B: "hai. watashi ga genki desu."
All the subjects and objects are there, but just not expressed but understood. But translating that omitted version isn't the literal translation of the whole sentence. It's merely the literal translation of the abbreviated sentence. I read in one book ("What the textbooks don't tell you" I think it was) a fine chapter about this "omitted subjects" and that newbie japanese learners often fall for the wrong(!) impression, that japanese has sentences without subjects. That's just wring. EVERY japanese sentence has it's subject, even if omitted. And that's the main part, why your translation ALSO should have it's subjects all in it's correct spot. As long as your native language isn't also in the habit of omitting subjects.
Thank you very much for great insights, everyone! Chris-san, you're right; if anyone can actually focus on how Japanese language works without relying on English too much, that's the best. However, this is often very difficult and many learners refuse to understand without translations. There are many Japanese learners who relies too much on English and always try to creat Japanese sentences translating every single one word from English. This also happens to Japanese people studying English, or anyone studying foreign languages.
Considering the fact that Japanese is a very unique language and super different from English, literal translation (I usually use "literal conversion" without the word "translation") doesn't often work. So, I think you've already found the very important point in learning Japanese!
The important thing would be to understand how this language works, with helps of English, and understand the grammar patterns. Good luck!