First mate, you lose my sympathy vote as soon as you said the "my legs are as strong.." etc bit. I'm sorry, but squatting is going to give you a bigger chest than chest presses will, that's my firm opinion and I've found it to be true.
I'll also tell you that isolating a particular muscle group (pecs in your case) will seriously stunt your pec growth. The human body is a unit, and for it to grow properly it has to be worked out as a unit.
And yes, you'll find people with huge upper bodies and underdeveloped legs - they wasted years just training their arms and chest when they could've got the same effect in half the time just by getting in that bloody squat rack and/or pounding the old deadlifts. But no-one wants to do those because they hurt. Yet it's proven time and time again that it's squats and deadlifts which make your biceps grow, not bicep curls.
Enough of my rant, I'm a bit pissed off today, I nearly picked up some pretty boy beach muscle buck fifty-three poser in the gym today and deadlifted him because he was doing bicep curls, just bicep curls, changing the weight every time, and looking as if he was humping the dumbbell rack whenever he tried to swing the weight up.
ANYWAY. I know how it is to not be able to afford a gym membership, I'd always say it was worth it if you can but I know that finances can be difficult .. I don't believe that you'll achieve the same level of growth with just home weights (since the weight you lift ultimately determines your eventual size gain) but yes, you can improve yourself.. but it's going to be a limited improvement.
Dumb-bells will only get you so far, and possibly nowhere at all. Until you get squatting, barbell rowing and deadlifting you can expect minimal growth, pure and simple.
If adjustable weights are your only option, and I do mean as a last resort - that buying a power rack is out of the question that is - then go for them, and best of luck. There's members here including myself who can tell you the right exercises etc, but I implore you to reconsider.
If it's really out of the question and dumbbells are the only thing then like I said, they're the only option. Otherwise you'd be best off saving up for a power rack and barbells, a home gym setup of sorts.
You really can't train back or legs with dumbbells, and that's what's gonna give you a huge chest, and avoid the chicken leg syndrome (and don't tell me your legs are big enough already, because that's what everyone says).
Have I been harsh? Probably, and I apologise, it was the leg comment that did it.
Good luck bro, I'll answer any exercise/form/weight-related question you like and advise as best I can, but you need a much better setup for serious growth. Save up for a few months.. once you buy your stuff, it'll last you ages.