Step 1: Finding the oxidation state of platinum.
Ammine ($NH_3$) is a neutral ligand and chlorido (Cl) carries a charge of $-1$ each. Since the whole complex $[Pt(NH_3)_2Cl_2]$ has zero overall charge, letting $x$ be the oxidation state of Pt gives \[ x + 2(0) + 2(-1) = 0 \Rightarrow x = +2 \] so platinum sits in oxidation state (II) here.
Step 2: Ordering the ligand names.
IUPAC naming lists ligands alphabetically by their name, not by charge or size. Ammine starts with 'a' and chlorido starts with 'c', so ammine is named first. With two of each present, these become diammine and dichlorido.
Step 3: Assembling the full name.
Putting the ligand names together in that alphabetical order, followed by the metal name and its oxidation state in Roman numerals, gives diamminedichloridoplatinum, with (II) written in brackets right after.
Step 4: Ruling out the other options.
Any name starting with 'dichlorido' before 'diammine' breaks the alphabetical rule, and any name ending in (IV) does not match the $+2$ oxidation state worked out above. \[ \boxed{\text{diamminedichloridoplatinum (II)}} \]