If you have access to dumbbell set, then you will be able to perform some good dumbbell exercises for developing and toning the chest. However, in my opinion it is much less necessary to use dumbbells to get an excellent chest workout over other muscle groups like the shoulders and arms as I believe the bread and butter of any chest workout is Bench Presses with a barbell.
Nevertheless, the following exercises work best with an adjustable dumbbell set such as the Bowflex SelectTech 552 Dumbbell set where you have access to all the different weight increments. If you have just a few dumbbells at home, some of the exercises may have limited use if the weights you have are too heavy or too light for them.
Dumbbell Bench Presses. Lay down on a bench and take a dumbbell in each hand. With the starting position of your hands at chest level with your elbows bent and forearms perpendicular to the ground, press each weight up until your arms are just shy of being straight and come back down. Do not lock your elbows on the way up as that takes work away from your muscles. Repeat for 8-10 reps.
Although Dumbbell Bench Presses can be a beneficial part of your workout as you can get more of a stretch in your chest than with Barbell Bench Presses, read my article The Advantages of Using Barbells over Dumbbells for Your Workout Routine for reasons why using a barbell is superior to using dumbbells for bench presses. If you only have dumbbells, then doing Bench Presses with them is fine, just don't expect to be able to do maximum weight with them.
Incline/Decline Dumbbell Bench Presses. These are the same as regular bench presses except the angle of the bench is changed. Move the bench to an incline while doing bench presses and you'll work the top of your pectorals and shoulders. Move the bench to a decline position and you'll work the bottom part of your pectorals. These exercises are great for ensuring a fuller chest and weak point training. If one half of your chest is lagging the other in size, you can focus more on the appropriate exercise.
Dumbbell Flys. This exercise can only be performed with dumbbells and it is an excellent one for really making your chest burn, particularly the inner half. Lie down on a bench with a weight in each hand. With your arms held out to the side and slightly bent as if you were about to give someone a bear hug, lift the weights upward and together by clinching your shoulders. Your arms should be locked in the same slightly curved position as they were at the start of the movement.
A lot of people like to clink the weights together and make a noise with them at the top during each rep. I find this dumb as for one, you're risking damaging your weights and two, there's no benefit to going through the full range of motion like that. By the time your hands are together straight up in front of you, you'll notice the pressure shifts from your chest to your shoulders. If your intention is to solely work out your chest during this exercise, I recommend stopping just a couple inches short of having the weights collide and then go back down in a fluid motion. I recommend 10-12 reps for this exercise.
Pullovers. Lie across a bench instead of along it with your hips dipped towards the floor, and arms above your head on the opposite side of the bench, stretching your rib cage area. Hold on to one fairly heavy dumbbell with both of your hands at one of the plated ends instead of the handle and lower the weight beneath your head then press back up so that the weight is in front of you again and repeat. At the bottom of each rep you should really feel a stretch in your chest, rib cage and torso.
If you are interested in purchasing workout equipment for your home gym, Amazon has some great products available.
If you are interested in more dumbbell exercises refer to:
How to build big Biceps
How to build big Triceps
How to build big Forearms
How to build big Shoulders
How to build a big Chest
How to build a big Back
How to build big Legs
How to build big Calves
More of my workout advice:
Are Bowflex SelectTech Adjustable Dumbbells suitable for you?
Should you buy Bowflex SelectTech Dumbbells in a store or online?
Why you would want to use barbells for your workouts
Why you would want to use dumbbells for your workouts
How I lost over 40 pounds in 6 months with some simple diet changes
Upper body exercises with no equipment needed
Lower body exercises with no equipment needed
Cardio exercises with no equipment needed
The Different Ways of Doing Dumbbell Bicep Curls
Inflatable Exercise Ball Exercises for Men