TL;DR
Taking into consideration that there are no direct answers about this by any of the people in charge of the movie or any in universe explanation about it. But lets consider some all-round knowledge, mainly the comics and Greek Mythology. Some of it you will have to accept that the comics already are very different from the Mythology since they have their own universe.
- The lightning could either be weather harness or actually Zeus' thunderbolt, since he is dead.
- Ares is able to have telekinetic control over every weapon in the comics.
- The killing of every god was a DCEU (DC Extended Universe) choice, which is weird.
- DCEU are the movies produced by DC starting with Man of Steel, 2013
LONG ANSWER
1. The Lightning
In my point of view, there are 2 main possibilities to go on WHY and HOW he was able to harness lightning:
- The comic book explanation
According to Ares's profile, he has many abilities other than just fighting, such as Energy Projection and Weather Manipulation. In my opinion, he could just have harnessed the natural lightning, and not used Zeus's signature weapon. (If I recall, the weather changed during there fight also. Correct me if I'im wrong.) Also, this could be an Ares thing that he used exactly to mock Zeus (who is dead).
Here is the origin of Zeus Power from his birth story:
After reaching manhood, Zeus forced Cronus to disgorge first the stone
(which was set down at Pytho under the glens of Parnassus to be a sign
to mortal men, the Omphalos) then his siblings in reverse order of
swallowing. In some versions, Metis gave Cronus an emetic to force him
to disgorge the babies, or Zeus cut Cronus's stomach open. Then Zeus
released the brothers of Cronus, the Gigantes, the Hecatonchires and
the Cyclopes, from their dungeon in Tartarus, killing their guard,
Campe.
As a token of their appreciation, the Cyclopes gave him thunder and
the thunderbolt, or lightning, which had previously been hidden by
Gaia.
Together, Zeus, his brothers and sisters, along with the Gigantes,
Hecatonchires and Cyclopes overthrew Cronus and the other Titans, in
the combat called the Titanomachy. The defeated Titans were then cast
into a shadowy underworld region known as Tartarus. Atlas, one of the
titans who fought against Zeus, was punished by having to hold up the
sky.
(My emphasis)
So now we may assume a few things by this. We know that Zeus' signature thunderbolt is his weapon, not an inherent power. In this particular DCEU, Zeus is dead and Ares was the only god alive of the Greek Pantheon for a while. Isn't it capable that he obtained Zeus' thunderbolt during this time?
2. Telekinetic Abilities
This, as explained above, actually is very capable that he would use. As described also in his profile, Ares in the comics has the telekinetic power over every weapon (What defines a weapon is really vague). Also, telekinesis is actually a very common power for powerful beings in movies.
3. Killed All The Gods
This was an DCEU choice. This does not happen in Greek Mythology neither the Comics. Actually Wonder Woman usually fights other Greek Gods such as Circe. I really don't know what they were thinking.
