我现在在看着一部巴西电视剧。我觉得故事很有意思。

现在在 is present tense of the continuous action or situation. I noticed there is an alternative way to express this, which is "正在" construction (progressive aspect) followed by the verb + 着. You have written this perfectly and also used a classifier to identify the noun. 

In English, I guess we have a short from of saying things quickly. So 现在 + 看一部巴西电视剧 is perfectly fine, as you omitting a construction of 正在 and 着.  In addition, 我现在正在看着一部巴西电视剧 is also grammatical okay.

The previous corrections are okay, but of course, this is for writing. If you wish to say it in speaking, then there'll be several adjustments needed to be made. Overall, your Chinese level is pretty good already, and it looks like you have put a lot of thought into understanding it too.

Good job!