Can the quality of your guitar hold you from being the best guitar player you can be? The answer is sort of tricky because it could be yes and no! If you are a beginner, you certainly don't need a Les Paul Custom and a Marshall Stack to become a great guitar player. On the other hand, if you are a seasoned guitar player and were handed a very cheap guitar and cheap amp to perform your gig with, it would be safe to say that you wouldn't play as good or sound as good as you're capable of. They would still be able to play the cheap instrument for all it's worth though!
With that being said, it's important to understand that when you are learning guitar, it's all about putting the hours in and really understanding what you are practicing. Then you actually need to spend quality time practicing those things on a regular basis. It's common to hear a guitarist in the early stages of practicing say something like "if I only had this guitar and that amp I would be able to play better." Not necessarily true. What you need is a guitar that plays good, and an amp that sounds good. Period.
This could be a fifty dollar guitar and a fifty dollar amp. You may need to make some minor adjustments to the guitar to make it play easier, and set the intonation, but after that, it comes down to you putting in the hard work. There will come a time in your guitar playing that a better guitar and a better amp will make you sound better. But... that's down the road a ways.
Things you can do to a cheap guitar to make it sound better are:
- Adjust the neck for proper relief
- Set intonation on the bridge
- Adjust string height
- file any frets that "fret out"
- New strings
Just these simple things can make a world of difference on an in-expensive guitar. If you want to take it a step further you could install new pickups as well. but, I would spend too much money, because eventually you will spend your money on a better guitar.
Keep in mind that a distortion pedal placed before almost any amp can get the job done if the amplifier you have doesn't have enough distortion for you. I recommend that you don't use too much distortion, but just enough to make it roar.
Remember this, you should be able to pick up an acoustic guitar and play it almost as good as your electric. If that is problem, then your electric guitar skills need to be refined. When you can pick up an acoustic guitar and play most everything on it that you can on your electric, then you're on the right path.