You can add data to the actual data structure by appending it to the employees array like
dataobj.employees.push({"firstName":$('input[name=firstn]').val(),
                        "lastName":$('input[name=lastn]').val() });
Of course, this requires that the JSON was parsed into the dataobj in the first place. It must be serialized again if you want to send it by GET. But it can be POSTed directly as a data object!
You can of course also start with an empty array, initializing dataobj like
var dataobj={ employee: [] };
before the above uptdating command comes into action.
A very late edit ...
Just in case there should be multiple firstname / lastname input fields, then the following will do a "much better" job (as it will have a look at all fields and collect only those where at least one of the names is set):
var dataobj={employees:[]};
function shw(){
  $('#out').text(JSON.stringify(dataobj).replace(/{/g,'\n{'));}
$(function(){
  $('#clr').click(function(){dataobj.employees=[];shw()});
  $('#go').click(function(){
   var ln=$('input[name=lastn]').toArray();     // ln: JS-Array of DOM elements
   $('input[name=firstn]').each(function(i,fn){ // for each fn DOM-element ...
    var f=fn.value,l=ln[i].value;               // get values as strings
    if (f+l>'') dataobj.employees.push({firstName:f,lastName:l}); // push name object
   });shw();})
  shw();
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<input type="text" name="firstn" value="John"><input type="text" name="lastn" value="Doe"><br>
<input type="text" name="firstn" value="Anna"><input type="text" name="lastn" value="Smith"><br>
<input type="text" name="firstn" value="Peter"><input type="text" name="lastn" value="Jones">
<input type="button" id="go" value="append names">
<input type="button" id="clr" value="clear">
<pre id="out"></pre>