How important is it to eat right and also..

"How important is it to change your diet if you are working to build muscle."
Depends on what your current diet is like. Might need change, might not.
Plenty of people add muscle with very ordinary diets as it's your training that drives the process.
If you have regular fitness, strength, physique goals a regular diet will work.

"and also can you gain muscle just from doing push ups and diamond pushups"
If doing them places enough stress on your current muscles to cause adaptation.
The problem with push ups is that you can't use increasing weight to progressively increase the difficulty and have to use volume which isn't as efficient and will most likely stop working for hypertrophy fairly quickly , depending where you are starting from.
For an extreme example after not being able to put any weight on one leg for 3 months just learning to walk again was enough to build a little muscle but that was very short-lived and I had to increase the stimulus (stairs, cycling, weights....) to continue rebuilding my leg muscle - just walking more wasn't going to do much.

Is that all you are planning on doing? What about the rest of your body?

funfan55 wrote: »

I plan to also build legs too and abs

Consider joining a gym to get the resistance you need to help you along.

A.C.E. Certified Personal and Group Fitness Trainer
IDEA Fitness member
Kickboxing Certified Instructor
Been in fitness for 30 years and have studied kinesiology and nutrition

Re: push-ups - a lot depends where you are right now in your training. As a completely unfit 50 year old woman, I went from no hint of muscle in my chest or arms to nicely defined chest and arm muscles in about a year of mainly pushups. I’m not huge, but as an older woman I get occasional “wow your arms and back are amazing” comments, so something is working. But I was starting from not even being able to do countertop pushups - I was so out of shape I was doing wall pushups - and working up to finishing the “bring Sally up” pushup challenge by the end of it. If you are a unfit noodle person you will get more gains from pushups than a big buff guy can.

A push-up is basically a bench press, inverted, so you can keep building muscle about as long as your bench press is less than your bodyweight - or I guess you could do what @SnifterPug suggested and add weight to your push-up. It may not be the optimal way to do it though, and eventually you are going to want to add weights to keep going.