You need to do much more than that. And yes, eat healthier and more to gain weight, that gives you a better chance at building muscle and not as much fat.
If all you have access to is bodyweight exercises, here's a decent home schedule:
Pushups and dips / core
Squats, lunges, and calve raises
Pullups / core
Rest one day. So that's three days of working out and one rest day then repeating. Don't do all those exercises in one day, it's meant to be pushups/etc Monday, then Tuesday for legs, Wednesday for pullups, Thursday rest, then repeat. Just example dates, you don't have to make it that.
How I'd set it up for a beginner:
Keep all exercises between 10-20 reps until you get stronger, and as for the sets
10 sets of pushups (wide for more chest), 5 sets of dips
10 sets of squats, 5 sets of lunges, 5 sets of calve raises
10 sets of pullups
And keep rest time between 30-60 seconds.
If you can't do pullups, aim for negatives on them. You don't necessarily need a pullup bar, just something that can replicate it, like a door, tree branch, or park stuff. What I mean by negative is jumping up so you're at the end position and slowly lower yourself.
Make sure to do ab/core exercises two-three times a week, those are important. Stuff like planks and crunches. Every other day/two days should be enough.
Of course there are probably more bodyweight exercises you can do, but those are the ones I can come up with off the top of my head.