Antoine Lavoisier was the first person to attempt to group the elements into 4 categories - gases, non-metals, metals and earths.
Johann Döbereiner realised that the elements with similar properties come in threes. He also saw that you could figure out the properties of the middle element by using the outside elements and their properties.
Alexandre-Emile Béguyer de Chancourtois recorded the atomic weights and put the elements into a system that showed the weight increase by 16 every rotation of the cylinder with all the elements with similar properties in a straight line.
John Newlands realised the elements that had similar properties had atomic numbers that differed by 7 (later changed to 8 after the discovery of noble gases). He did not leave gas for undiscovered elements, forcing him to fit two elements into a box for the pattern to continue, this is why he was not recognised and his paper not published.
Julius Lothar Meyer made the periodic table with the elements organised by weight and then vertically with elements with the same valency. This work was not published until 1870. He also showed how the atomic weights and the atomic volume made trends in the elements.
Dmitri Mendeleev created the current periodic table, identical to Meyers although they were unaware at the time. Mendeleev's system was the most successful as he left gaps for undiscovered elements, he managed to figure out the properties of five elements after seeing what they needed to be to fit in the pattern. He also did not put all the elements in with the right atomic weight but instead swapped them around so they would be in their groups and fit the pattern as well.
Henry Moseley used an X-ray to find out the atomic mass of the elements which confirmed Mendeleev's theory after recording the wavelengths and seeing the square root of the frequency made a straight line with the atomic number. He also noted the higher the amount of protons, the more energy was given out from the wavelengths.