How can you tell where an element will be in the periodic table and how reactive it will be from its electrons?

If you assume the atom is not ionized, then yes, you can deduce the location on the periodic table and the reactivity. However, electrons are extremely fickle.
If an atom is not ionized, it will have the same number of electrons as protons. Therefore, counting the electrons will give you the atomic number, which corresponds with its location on the periodic table. Additionally, you can use this number to estimate reactivity (or electronegativity), which is a function of how likely an atom is to bond with other atoms.
Generally speaking, electrons sit in orbitals, which contain specific numbers of electrons in them. The closer an orbital is to being full or empty, the more likely it is to react something. (Most orbitals can have a maximum of 8 electrons. Carbon has 6 electrons, which means it is more likely to accept electrons than it is to give them away. Hydrogen, on the other hand, has 1 electron, so it gives it away easily to form a bond.) This is just a general rule of thumb, because the laws of particle attraction and bonding can get incredibly complex. However, noble gases are atoms that have filled up their orbitals, and therefore do not react (i.e., do not give or receive electrons).
So, assuming the number of electrons is unchanged, yes you can. However, electrons can be added or taken away very easily and still be a part of the same atom. If the number of protons changes, it is by definition a different element. Therefore, the atomic number, which counts protons, is used to order the periodic table.