Question: I have read both the styles...

> 
> Neither I like you, nor do I like your family  
> 
> **and**  
> 
> 
> 
> 

Is there any special cases where we use just 'nor' instead of 'neither...nor'? Has it to do anything with emphasis?

Answer: 
In some cases, we may use just "nor" which suffices to replace both neither and nor such as "He nor I was there", but it is not only archaic but also sounds awkward. We may also use nor to replace the preceding neither such as "Nor he nor I was there". That's also archaic. I don't think such a use fits in the structure of modern English. (Please refer to The Free Dictionary).
In my opinion, the sentence "Neither I like you, nor do I like your family" should begin with an inversion, like so:
Neither **do I like** you, nor do I like your family. This is the case when we link two clauses.  

Alternatively you could write:
"I like neither you nor your family", this time without a comma before nor.