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: 
As far as I know, we use the word NOR as an additional "information/support/reason" to a preceding "negative(usually)" statement. It should come after the "negative(or positive)" statement.

Ex: 

1. I **don't** like you, **nor** them.
2. **Neither** you **nor** I will go to the party!

It would be awkward to say:

1. You like him, nor I.
2. I will go to the party, nor you.

Bottomline? We use nor as an **additional** reason/support/information to a preceding statement, usually a negative statement.
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.