Know the Organisation’s History
Most organisations have a pattern of hiring and dealing with personnel. Study this pattern and see what they have been classically offering their employees and if there is a difference in what was offered to you. This will give you an idea if they are likely going to grant you your wish or not.
Arrange a Meeting
Ideally arrange a meeting in order to discuss the offer that you have received. Meeting face to face can be great in this regard and you can present your side of the matter in person. You can email them in a professional manner or even write them a letter but a personal meeting is still a better option.
Be sure that whichever way you communicate your counter offer, you are polite. Being energetic is fine but do not be erratic. Even if you feel that the offer is unjust and you are definitely qualified for the job, you should still stay calm and discuss the matter. Be convincing and do not be pushy as not all human resource managers will be understanding or caring. Also make sure that you listen to the other party’s point of view as well.
Know the Market Situation
Be sure to know what the going rate for the position is that you have received the offer for. This can be a big help and you can negotiate on the basis of what is being offered in the market as opposed to what you have been offered by that organisation.