How do I get the back off an Accutime model #TMNB003 Ninja Turtles Watch?

Your model returns no results. Did you buy it on Amazon or eBay? They are all copies made in China.

https://www.google.com/search?q=accutime+watch+kac3050cp

From their website

https://www.accutimewatch.com/instructions

..

I do not find instructions for the watch at the Accutime website, via Google, nor at YouTube. I do find two contact numbers for Accutime Corporation:

toll free number: 855 222 8846
The above number appears associated with the T5 tracker model but one hopes support extends beyond such.

The below number is not toll free but is also associated with Accutime Corporation in New York.

212.686.9220

It is bust it is just rubbish made and it is bust

Try looking between the two buttons on the right. There's a tiny knit-sized button between them that's barely noticable. Ourc watch model is PAW4307ds, but it is likely the same configuration.

Chances are the plastic piece around the movement of the watch broke or is out of place. I suggest taking it to the jewelry store and have them correct this for you. It should not be too much money to fix this. The other option is to pop the caseback open to see if the plastic piece is stil in place if not then you can easily put in place again. However be carefull not to scratch the caseback and sometimes it's hard to insert the caseback on again. You would probably need a tool.

You need to have case back opener. look for this site to know what it is look like. http://www.tztoolshop.com/FB_Catalog_Case_Back_Openers.html then if you have these tools, all you had to do is to put exactly the three pointed metal on the holes in your caseback and turn it counter clockwise. you need force to do this.

Some of the inexpensive watches like to mimic the expensive watches and put fake dials on the face of the watch. That way you get a Breitling look alike at the price of a Timex. If there don't appear to be controls for the dials, chances are that they aren't real dials, Look closely.

No, there is nothing wrong with the alarm in your watch.
The sound in all watches are generated by passing vibrations to the metal tweeter which is connected to the watch caseback inside the watch. As the surface of the watch is quite small, there is not efficient power to spred the vibrations over the caseback surface, hence low volume of the alarm. No wristwatch in the world is capable of loud alarm sound yet. This is something the scientists have to work with.
The plastic protector is definetly affecting the volume. Take it off and you will notice slightly better sound. This protector actually is ment to protect watch caseback from scratches prior to selling it to the customer and have no use after that. In fact it only collects the dirt and bacteria and can affect your skin as the caseback as well.
Another sound improvement can be the new battery if the original is underpowered. Alarm feature uses the most of the energy stored in battery and the power consumption is higher than all watch movement parts put together.

Rate me, please.

For closing your caseback you need the man with very strong hands first, but it might not work as some casebacks can be closed using special tools only.
Before you visit the watchmaker, try this: first of all open your bracelet by taking out spring lug. This will make more space at the back. Then take watch caseback and check for groove in it. Align that groove with stem (pin going into watch movement) and deepen grooved caseback side as it should be. Then without turning caseback apply great pressure with both hands to the watch(from caseback and dial sides). If hands are strong enough, caseback should snap on into place.
In case if it doesn't work, find a piece of thick leather (wide mens belt will do) and place it on completely flat surface (table top). Place the watch face down on that leather, adjust caseback groove with stem again and try to press caseback down with both hands, only this time using your weight.
If all mentioned above doesn't work- you have to go to see watchmaker.

Good luck.

Arthur