There is no way to earn Platinum directly from missions. The only options are to buy it from DE or trade items with other players for it. Items that will give you the largest amounts of platinum: Rivens, new mods, new Primes and vaulted Primes.
So here’s how to farm Platinum through selling Primes:
● Create an account on Warframe.market.
● Farm relics.
● Open the relics through Fissures (Capture / Exterminate / Interception missions recommended for speed).
● Use public squads so that you still have a chance at a good reward in case your own relic gives you something of low value.
● List the item on Warframe.market. The lower the price, the faster it’ll sell.
● You can also check what the latest mods are and try to farm them, though this can also be less efficient due to their drop chances (or due to various standing limits).
● Play the sortie to try to get a Riven mod. Then either sell it through Riven.market, or re-roll it with Kuva for better stats first and then sell it. This can get you a much larger amount of Platinum than primes or mods, but it can also take much more time due to the RNG or re-rolling the stats and farming the Kuva.