Initially, Democritus, a Greek in ~400 B.C. coined the term atomos (becoming atoms later) to describe the "ultimate particle", or that it is what makes up matter, and cannot be made of smaller pieces. In that era however, they were not able to perform the kind of experiments needed to show evidence for their theories. Essentially, John Dalton gave science his theory of atoms from his publication "A New System of Chemical Philosophy" in 1808, and used evidence from observation to support his theory. I would say most chemists would give the credit to Dalton for establishing the characteristics pertaining to atoms.

The idea of having "atoms", or really tiny particles that make up everything, has been around for at least a couple thousand years, and is known to exist as early as ~600 BCE in Greek and Indian philosophy. However, our modern concept of atoms and matter really started at the beginning of the 1800's with John Dalton, who really came through with solid evidence that matter was made of atoms, and explained why reactions of elements always occur in fixed ratios, and that atoms of different elements could be combined together to make new compounds..

Scientific discoveries are inherently continuous (our modern view of the atom is still changing vis particle physics), but most would probably pin the tail on Dalton. His experiments demonstrated integer-ratio combinations of elements into molecules, strongly suggestive of what we think of as atoms today.

John Dalton sort of made it inescapable, because elements react with each-other in specific,quantized, ratios, which almost cinched that there would be atoms. Dimitri Mendeleev further drove the point home by working out atomic numbers and placing elements in related series. It was Avogadro who established how many there are.