While not as good as a real pronunciation coach, one thing that helps me go beyond just trying to replicate someone else making a particular sound until they go "yes, that's it", a technique that is frustrating at best, is the use of the International Phonetic Alphabet (IPA). The best resource for it would be https://en.wikipedia.org/wiki/International_Phonetic_Alphabet
The goal of the IPA is to provide an alphabet so linguists can describe the sounds of all natural languages on paper. As such, they needed to map the whole space of sounds that are believed to be feasible for humans to produce. As such, every sound is mapped to a symbol, sometimes with diacritics, and it is placed on a series of tables describing how it is produced. For instance, according to the Wiktionary, "king" is pronounced /kɪŋ/, or /kiŋ/ in some American pronunciations (think of the slashes in API as quotes for the general sound, and square brackets as quotes for the more specific sound). Looking up the character "ŋ" in the IPA "pulmonic consonants" chart tells me its manner of articulation is "nasal", pushing some air through the nose, and its place of articulation is "velar" or with the tongue touching the soft palate. Looking at the features of the "voiced velar nasal", Wikipedia lists more detail:
- Its manner of articulation is occlusive, which means it is produced by obstructing airflow in the vocal tract. Because the consonant is also nasal, the blocked airflow is redirected through the nose.
- Its place of articulation is velar, which means it is articulated with the back of the tongue (the dorsum) at the soft palate.
- Its phonation is voiced, which means the vocal cords vibrate during the articulation.
- It is a nasal consonant, which means air is allowed to escape through the nose, either exclusively (nasal stops) or in addition to through the mouth.
- It is a central consonant, which means it is produced by directing the airstream along the center of the tongue, rather than to the sides.
- The airstream mechanism is pulmonic, which means it is articulated by pushing air solely with the lungs and diaphragm, as in most sounds.
You can even find a few pictures in related articles showing the shape of the tongue in the mouth. That's pretty much the best you'll get aside from a personal coach. Some of these explanations make more sense when you've learned how the whole system was devised, but it's not a pre-requisite to improving your pronunciation of a few specific words: just look up the word in a dictionary that lists the pronunciation using the IPA and look up each component symbol on Wikipedia.
Sounds in a language are not perfect exact notes, rather they are contrasting with each-other. Why do speakers of French hear a difference between some sounds that you (probably) don't? It's because in French, they contrast with each-other, while they don't in the languages you speak. That distinction is often harder to recognize than it is to produce once you know what to do with your mouth. Once again, the IPA charts can help, if it says two vowels contrast because one is rounded and one is not, then alternate between saying the sound with your lips flat and your lips rounded. If you can't really hear the difference though, it doesn't necessarily mean others won't.