How would you begin this?

Here's a rough sketch of how it might look:

roundLength = 60

function getPlayersAlive()
    -- Look through all the players who aren't in a lobby and check if they
    -- are on the right team, etc,
    -- returning a list of players
end

function roundFindWinner()
    local startTime = tick() -- Current time
    while tick() - startTime < roundLength do
        wait(1)
        local peopleAlive = getPlayersAlive()
        if #peopleAlive == 1 then
            -- Last person standing.
            -- Different lists for different teams and checking that
            -- one of the lists is empty would work, too, depending
            -- on the rules of the game.
            return peopleAlive[1].Name
        elseif #peopleAlive == 0 then
            -- No one is left playing, so there was no winner
            return "none"
        end
    end
    -- Out of time.
    return "none"
end

while true do
    wait(5) -- Time between rounds
    local winner = roundFindWinner()
    if winner ~= "none" then
        -- `winner` is name of player who won!
    else
        -- no one won (out of time or no survivors)
    end
end