I want to learn the aviation alphabet, and when searching the Internet on how to, I see a plethora of approaches, so I thought of asking here!
If it's primarily opinion based, let me know.
Presuming you're referring to the ICAO Alphabet* , (alpha, bravo, charlie, delta, echo, etc.) the easiest way (in my experience, from learning it as a disambiguator for phone work) is repetition, same way you learned the alphabet as a child, or the same way you'd memorize a poem. Get a written list of the letter names, read them through several times a day, and over time, start reciting without reading.
Do this several times a day for a week, you'll be close if not there. A second week and you should start to think in phonetic alphabet. A third week and you'll wonder why everyone doesn't think in phonetic alphabet.
*corrected from comments, originally International Phonetic Alphabet, which is a whole different thing.
Study like it's schoolwork!
I've found that quizzing yourself in a similar way to studying for a test is the best way to learn. I've had to learn a few different alphabets for different reasons, but there's always good resources online. I'd recommend two tests that I always give to beginners.
Beginner Quiz: https://www.sporcle.com/games/g/alpha_en
Intermediate Quiz: https://abg.ninja/alphabet
Ignoring the childish UI, these are actually great ways. Begin with the first link, and type in all of the letters that you know. This will very quickly show you which ones you know, and which you don't. Once you've figured out the ones you don't know, check a chart, attempt to memorize them, and try the test again. Rinse and repeat until you've memorized it fully, and can do it an hour after looking at the chart (so the information isn't remembered solely for being fresh in your mind).
After you've done this, use the second link I sent you to improve your usage, and to string multiple letters together. Continue until you're comfortable.
It's incredibly helpful to read the letters out loud while doing these tests.
My favorite method for memorizing that kind of stuff is to create flash cards with a picture that can be associated with the word. You might have to use your imagination for find a suitable image for each letter, but once you do, the memorization efficiency goes way up.
Another option is to make flash cards with the letter and its phonetic word in a huge font so it takes up an entire page. You will find you are able to recall the "image" of the word more easily than just the information itself. I use that method for memorizing operating limitations where it's just a numerical value you have to store in your head.
I can recommend the (free) site http://radioalphabet.com/, specifically the Flash Cards tab. It provides you with a word, listed vertically with an input box besides each letter, and requires you to write the second and third letter of the word for each letter (which are unique amongst the words). It then automatically jumps to the next box, allowing you to practice quickly recalling the words for each letter.
After you have used it for a little, the Statistics tab provides you with hit and miss rates for each letter, so you can know which ones to focus on memorizing. I personally schedule repetitions in my calendar, using the principle of spaced repetition learning mentioned in insysions answer, so I increase the time until next repetition if I get everything right.
Make drawings (it doesn't matter how good you can draw). Use whatever comes to your mind first. For example: for Charly I think of Charly Brown. For Mike its another comic figure called Mike, but unlike Charly he is very tall, so these two side by side look very funny. They together (funny view remember) are visiting Quebec in November...
... where they meet Romeo and Juliet who are dancing Tango.
Take just a few at a time. Make each of them tell a little story. Stick them where you can see them several times a day. Bathroom for example.
If your brain has something to visualize its more likely that you will remember.
I had the same problem. The easiest way to me to remember all the letters is with the car plates. It's fun and also keeps you practice the alphabet without a specific order, that's important too.
Spaced repetition learning. E.g. https://www.memrise.com/course/31682/nato-alphabet-3/