Emptying and populating an element in jQuery
Your code isn't bad, but there are some aspects that tick me a bit.


Lets start with the generateRandom()

You currently have the following:

var objCount = Object.keys(cards).length - 1;
var number   = Math.floor(Math.random() * (objCount - 0 + 1)) + 0;

return number;


And I ask:


Why + 0 or - 0?
Why Object.keys(cards).length - 1; for an array?
Why take 1 to sum it again?
Why that bloat?


Here's how I would write it:

function generateRandom()
{
    return Math.floor(Math.random() * cards.length);
}


Since the length is 1-based and the indexes are 0-based and Math.random() returns numbers between 0 and 1 (excluding the 1), you don't need more than this.
And now, the drawHumanCard()

You have the following code:

var random = generateRandom();

var humanEl = $('#human');

humanEl.find('#human-name').empty().text(cards[random]['name']);
humanEl.find('#human-job').empty().text(cards[random]['job']);
humanEl.find('#human-image').attr('src', 'img\/trumps\/' + cards[random]['image']);


You can store cards[random] inside a variable, like this:

var card = cards[random];


And to improve performance, you don't need the .empty(). It's a waste of time and CPU power.

Also, why humanEl.find('#human-name') instead of $('#human-name')? And why not an array with these objects?

I would rewrite the whole thing as this:

var human;

//it must run after the elements exist in the DOM
$(document).ready(function(){
    human = {
        name: $('#human-name'),
        job: $('#human-job'),
        image: $('#human-image')
    };
});

function drawHumanCard()
{
    var random = generateRandom();

    var card = cards[random];

    human.name.text(card['name']);
    human.job.text(card['job']);
    human.image.attr('src', 'img/trumps/' + card['image']);
}



So, the final code would be like this:

var human;

$(document).ready(function(){
    human = {
        name: $('#human-name'),
        job: $('#human-job'),
        image: $('#human-image')
    };
});

var cards = [
    new TopTrumpCard('Name 1', 'designer', 'image1.jpg'),
    new TopTrumpCard('Name 2', 'designer', 'image2.jpg'),
    new TopTrumpCard('Name 3', 'designer', 'image3.jpg')
];

function TopTrumpCard(name, job, image) 
{
    this.name = name;
    this.job = job;
    this.image = image;
}

function drawHumanCard()
{
    var random = generateRandom();

    var card = cards[random];

    human.name.text(card['name']);
    human.job.text(card['job']);
    human.image.attr('src', 'img/trumps/' + card['image']);
}

function generateRandom()
{
    return Math.floor(Math.random() * cards.length);
}


This can and should be written as vanilla Javascript. The use of jQuery isn't justifiable here but that is an exercise for the O.P. to try.