At least as I understand it, de Broglie more or less had the idea, but as with many things in physics, it wasn't put on rigorous mathematical footing right away. What matters ultimately in science is what nature does, you can speculate all day until you're blue in the face, but at the end of the day you have to check against nature. And wouldn't you know it, at about the same time an experiment was done to check whether de Broglie was right, not only that matter behaves as a wave, but also that his wavelength was right. https://en.wikipedia.org/wiki/Davisson%E2%80%93Germer_experiment
The experiment was essentially firing electrons with a specific momentum at a nickel target and measuring the interference pattern caused by scattering in the nickel. It's essentially a more technical version of the Young double slit experiment, which Akira Tonamura performed the electron version of in 1989 https://www.youtube.com/watch?v=jvO0P5-SMxk . If you read up on the Young experiment, you will see a formula $d\sin{\theta} = m\lambda$. Using this, you can determine the wavelength by measuring the deflection angle of each of the fringes if you know the distance between the two slits that the wave passes through.

I hope this helps, cheers!

Edit in response to comment: The fields that particles belong to are described in Quantum Field Theory. If you want an introduction to ideas involved, I found Griffiths' Introduction to Elementary Particle Physics to be helpful. There really is an electron field, but in the context of QFT, we usually only calculate how much particles scatter off each other, not so much how they freely propagate in space. To properly understand what's going on, you also need to understand how photons can be a thing if light is a wave: https://en.wikipedia.org/wiki/Quantization_of_the_electromagnetic_field

As for the second part, $E=pc$ is a special case of $E^2=(mc^2)^2 + (pc)^2$, which $E=mc^2$ is a special case of as well, for massive particles in their rest frame. This comes from special relativity, which already existed in 1924 when de Broglie posited his idea. Also, in actuality $E=hf$, it just so happens that since the frequency of light is $f=\frac{c}{\lambda}$. The equation for light that he used was not for energy, it was the one for momentum $p=\frac{h}{\lambda}$. He essentially said that perhaps this would work for massive particles too, using $pc=\sqrt{E^2-(mc^2)^2}$. If you're doing this with electrons in a cathode ray tube, you can get the energy of the electrons by adding the mass energy to the kinetic energy given to the electrons by accelerating through the electron gun.