Here's what you should already have:

A FitBit account which should be synced 3+ days prior and make sure the account isn't a really old one from a few years ago.

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Where to contact:

Once you've logged in you can navigate to the Help section and click the "Get Support" button

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You can use the Live Chat - Email - Call but I always use Calls.

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What the fuck do I say:

Below a Live Chat Transcript (with some things cut out for privacy reasons) which you can use to help you (This also works for calling but you have to be good at talking to real humans!)

Isabel: Hi Scott, my name is Isabel. How can I help you?
Me: Hi Isabel, sorry to bother you but I'm having an issue with my Charge HR
Isabel: I will be happy to find a solution for your Scott.
Isabel: Can you please tell me more about the issue?
Me: Thanks, basically it wont take Charge - I think the battery is broken
Isabel: Thanks for the details, in order for me to further investigate the tracker's behavior, I'd like to know the email associated with it
Me: Okay, its (The account's email here)
Isabel: I appreciate that, bare with me
Me: Thank you
Isabel: In the meantime, I'd like to know if you have tried to troubleshoot this issue
Me: I've read a few guides on how to troubleshoot it but nothing's worked so far
Isabel: Can you please tell me what have you tried so far?
Me: The charging pins are clean
Me: I've made sure its securely plugged in
Me: But nothings seemed to work
Me: So I had no option but to turn to support from you guys
Isabel: I appreciate your efforts to fix the tracker before contacting us.
Isabel: I was able to confirm on my system, that your tracker's last sync was on May 9th, with a 5% battery level
Me: Yes
Me: Since then it won't take charge
Me: Which is baffling since it worked flawlessly prior to this
Isabel: I would recommend to restart your tracker.
Isabel: Can we do that?
Me: Sure I can try that
Isabel: Perfect!
Isabel: Can you please connect your tracker with the charging cable?
Me: Alright, bare with me
Isabel: Thanks
Me: Done, what next?
Isabel: While is connected, please press and hold the button for 15 seconds
Me: Will try now
Isabel: You should be able to see a Fitbit Icon and a number
Me: Sorry, there's no feedback coming from the device
Me: Any idea as to why it's not responding?
Isabel: Most of the time is not about the battery, is the display not turning on and it can be easily fixed with a restart
Me: Mine must be an unlucky exception, I've tried again but to no avail
Isabel: No problem, I appreciate your cooperation
Isabel: Since the tracker is not responding to our troubleshooting. I'd like you to keep in mind that Fitbit stands by it's products and I'd like to to to the next step
Me: Okay, as long as I can get back on track with my fitness tracking
Isabel: Of course! that's our main goal, to exhaust all the options to keep you enjoying the Fitbit experience.
Isabel: I'd like you to help me with the next information
Isabel: Where did you purchase your tracker?
Me: Sure thing
Me: My tracker was a gift from my son in law for my 53rd birthday
Me: He knew how keen I was at fitness and knew I'd love a FitBit of my own
Isabel: If you allow me, he is a really nice son in law and he gave you such a nice present.
Isabel: We want you to continue taking advantage of this gift
Me: Thank you very much
Me: I'm sure he'd be very happy to hear that!
Isabel: Scott, I have one more question, what's the country where you currently live?
Me: I am staying in Ireland currently
Me: Lovely country, but very cold...
Isabel: I wish it was cold where I live.
Isabel: Scott, I'm really happy to let you know that your tracker is still covered by our two-year Fitbit warranty
Me:That's a relief
Isabel: I'll be glad to provide you with your replacement options!
Me: That's fantastic, hopefully I can be up and running soon
Isabel: It's nice to see that you like the outcome!
Isabel: I just sent you an email with your options, it was sent to the email associated with your tracker
Isabel: Can you please check your inbox?
Me: Sorry for the inconvenience but would you mind sending it to my business email?
Isabel: No problem, what's your business email?
Me: It's (email here)

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Pointers:

Start of nice, play it out like you're a friendly guy and mention things like "sorry to bother you"
Get to the point, tell him/her your tracker won't take charge for some reason and that you've tried to troubleshoot it. You can say things like "I've looked up some guides" or "I've tried to clean the charging pins"
He'll/She'll ask you to do some further troubleshooting and take 1-2 minute breaks after each task he/she asks you to do to make it look like you're genuine.
Tell him/her that it just wont work, I'd usually say "Thank you for the help but I've tried everything but to no avail"
He'll/She'll then tell you about your warranty options, an email will be sent with your options (get a replacement)
The issue is that he'll/she'll send it to the original owner's email inbox, so ask her to send it to another email (your email) and say it's a business email that you prefer using.
Once you get the email, you're all set. There'll be an option to get a replacement tracker and it's easy from there!