What were the United States' and the Soviet Union's roles in Lebanon during its civil war?

The Cold War was an indirect conflict.  That is, the
Soviet Union and the United States did not fight each other directly, as both sides had
nuclear weapons and that kind of conflict was bound to turn out badly for both sides. 
So they competed through other countries, and the proxy wars
there.

Assuming you mean the civil war of the 1980s, the US
role in Lebanon, however, was both direct and indirect.  Indirectly, the US was a strong
supporter of Israel (as it still is our most important ally in the region), and weapons
from the United States had made their way in foreign aid to Israeli forces and also to
the Israeli-backed Christian Druse militia fighting for contorl of Beirut and other
regions of the country.  So we indirectly armed one side in the
conflict.

Later, President Reagan ordered in Marines as
peacekeepers, and ordered naval gunfire to support the militia we backed in their
battles for control.  This was short lived on both counts, as a suicide bomber killed
263 Marines, after which we quickly withdrew.

The Soviet
Union had extensively armed Syria, which occupied the Bekaa Valley in the eastern half
of the country, and massacred 5000 Christians with Soviet made weapons and ammunition. 
Most Islamic militants and factions also used Soviet weaponry extensively during this
civil war. 

Lastly, both countries squared off against each
other in the United Nations, vetoing each other's proposals so that the UN could play no
meaningful role in ending the conflict.