First, select the element. You can usually use the ID like this:
$("#searchField"); // select element by using "#someid"
Then, to set the value, use .val("something")
as in:
$("#searchField").val("something"); // set the value
Note that you should only run this code when the element is available. The usual way to do this is:
$(document).ready(function() { // execute when everything is loaded
$("#searchField").val("something"); // set the value
});