Your readers will be able to tell if your characters are giving off real feelings, or just going through the motions. If you can't make the audience believe that Jimmy is heartbroken over the death of his parakeet, you need to rework that section. You need to not only have already given them reason to feel bad for Jimmy, as well as the parakeet, but you then have to be unafraid to really rip Jimmy apart when it happens and show his true colors, because when you're really and truly heartbroken or enraged about something, there's no hiding anything—no way in Hell.
How does it work, though? Think back through your life—we've all felt strong emotions. It's kind of the writer's bread and butter—if you haven't had some kind of meltdown at some point in your past, you're kind of just a fledgling writer. I am joking of course...a little. It's much easier to write if you've already been cut and sliced to bits and had your emotions laid out on the table at least once, multiple times being preferable. It really sucks, but that's the way these things work.
So you take that real emotion you've felt—not seen but actually felt—and you dive into it. You hurt yourself again or, if it's intense joy, you bring that feeling over yourself all over again. As soon as you start feeling with your character, that's when the reader will start feeling for your character.
If you've done this, you'll probably see a flaw right about now—how the hell are you supposed to write in this state? You can't even see, for Christ's sake! You have to either dial it back enough that you can type, but still keep the memory of that emotion fresh in your brain, or you have to hand-write/touch type your scene while your in the midst of that pain. It's the most effective way to make it all properly poignant.
Of course, you should also avoid saying the actual emotion in the scene. Jimmy's sad—don't tell us, because then we know what's coming. You have to show us without using sad, upset, depressed, or any other words like that. Talk about the cold in his limbs, the parakeet's silence, the churning of his lunch, the hot tears shredding down his face—but never say that Jimmy was sad unless you're writing a children's book...and I'd avoid it then, too, if at all possible.
But, more important than any of this (hopefully practical) advice is this: read it. If you feel the right way, you're at least headed in the right direction and, while you're not as impartial as the average reader, that outpouring of emotion from you will get you somewhere. Cross my heart and hope to die.
Now emote, my lovelies!