What are best ways for negotiating with HR when a salary bracket/slab offered to you is already less or below industry standard?

Firstly, remember that there is a lot more to compensation than just the salary.  Benefits, prospects, training - even the commute - all play a part.

Secondly, figure out your bottom line.  What is the point at which you will really say, no, I won't take this job.  This number is private and should never be disclosed to the other party.

Thirdly, try and figure out a deal that might work for both parties.  Is there something you can offer? 

Fourthly, arm yourself with as much information as you can.  Industry comparisons are useful.

Now you're ready to  negotiate.  Remember that you will never get more than you ask for.  If you have received a salary offer, reply to each bit of it in detail.    For example, you could say something like,

'I would love to work here and I'm happy with all the benefits, but the job involves long hours and I have young children.  One of my motivations for moving was to spend a bit more time with them, so I'm looking for5 weeks holiday rather than the4 you have suggested.    On the salary, my research (enclosed) suggests that the going rate for this kind of role is $X, and that's the number I have in mind.'

You might not get anywhere, but there is no harm in asking.  In general, the more senior you are the more likely you are to get a good response. 

Never disclose your previous salary to a new employer, unless it's to your advantage.  What you earned is none of their business.  Instead, mention your target salary.