  Two-dimensional (2D) magnetic materials hosting nontrivial topological states
are interesting for fundamental research as well as practical applications.
Recently, the topological state of 2D Weyl half-semimetal (WHS) was proposed,
which hosts fully spin polarized Weyl points robust against spin-orbit coupling
in a 2D ferromagnetic system, and single-layer PtCl$_3$ was predicted as a
platform for realizing this state. Here, we perform an extensive search of 2D
PtCl$_3$ structures, by using the particle swarm optimization technique and
density functional theory calculation. We show that the desired PtCl$_3$ phase
corresponds to the most stable one at its stoichiometry. The 2D structure also
possesses good thermal stability up to 600 K. We suggest SnS$_2$ as a substrate
for the growth of 2D PtCl$_3$, which has excellent lattice matching and
preserves the WHS state in PtCl$_3$. We find that uniaxial strains along the
zigzag direction maintain the WHS state, whereas small strains along the
armchair direction drives a topological phase transition from the WHS to a
quantum anomalous Hall (QAH) insulator phase. Furthermore, we study bilayer
PtCl$_3$ and show that the stacking configuration has strong impact on the
magnetism and the electronic band structure. Particularly, the $AA'$ stacked
bilayer PtCl$_3$ realizes an interesting topological state -- the 2D
antiferromagnetic mirror Chern insulator, which has a pair of topological
gapless edge bands. Our work provides guidance for the experimental realization
of 2D PtCl$_3$ and will facilitate the study of 2D magnetic topological states,
including WHS, QAH insulator, and magnetic mirror Chern insulator states.
