Hello, I am just wondering what the ruling about temporary hit points coming from two spells. Especially when a spell has an additional effect. Such as Armor of Agathys and False Life. 

Armor of Agathys (1st level) grant 5 temp hit points and will deal 5 cold damage to others who hit you.

False Life (1st level) grant 1d4+4 temp hit points.

Let's say I used Armor of Agathys and was hit a few times but still have 1 temp hit points. Now someone casts False Life on me .

1. Do I gain the temp hit points granted by False Life (not stacking)?

2. Do I retain the buff granted by Armor of Agathys still?

1. You get to choose which ones you keep "If you have temporary hit points and receive more of them, you decide whether to keep the ones you have or to gain the new ones. For example, if a spell grants you 12 temporary hit points when you already have 10, you can have 12 or 10, not 22."

2. Only if you choose to keep the 1 THP over the 1d4+4.

2. is a definite no if you have chosen to take False Life's temporary hitpoints - otherwise you could keep it almost indefinitely. The effects of Armor of Agathys only remain while you have the temporary hitpoints from it - not temporary hitpoints in general:

You gain 5 temporary hit points for the duration. If a creature hits you with a melee attack while you have these hit points, the creature takes 5 cold damage.

Temp HP, and any bonuses/effects derived from them, are mutually-exclusive discrete pools. If you have temp HP from one source, and are subsequently granted temp HP from a different source, you have to choose which one to use. If you accept the new temp HP from the subsequent source, it completely overwrites your old temp HP, and overwrites any bonuses/effects derived from the original source.

I.e., if you have Armor of Agathys temp HP, and accept new temp HP from False Life, you no longer have any temp HP or effects from AoA.

Much thanks for the help everyone

Unfortunately you cannot take advantage of other THP sources to keep Armor of Agathys on-going. The best way to do it is through protective abilities like Arcane Ward from Abjuration Wizard.

The Arcane Ward is so weird and complicated though, that you'll start provoking questions like whether your resistances apply to damage taken by the Arcane Ward, whether hits against its THP pool would even trigger your Armor of Agathys damage at all, etc. (For the record, I'd say yes to the first question probably, and yes to the second definitely, but the point being its a complicated class feature)

I agree, but by RAW it’s totally ok. Complication piece of Arcane Ward is the management efforts to restore the ward through Abjuration spells.

AoA is straightforward: if you got hit and still have the THP from the spell, enemy will receive the retributive cold damage. This is one the few spells out there that scales really well.

Arcane Ward is not a secret: every time you take damage, your Ward absorb that instead of you, considering it hit points. Any remaining damage goes to you — and any THP you might have.

In practice: let’s imagine an Abjuration Wizard 5 (that got AoA through Warlock dip or some other source like racial abilities from Mark of Warding Dwarf or Levistus Tiefling).

If you cast AoA with a 3rd spell slot, you will have 15 THP and you will give 15 cold damage every time you get hit in melee. As a level 5 Abjurer, your Arcane Ward has 13-14 hit points (let’s consider 13). If you manage to keep the 15 THP + 13 HP from Arcane Ward always on, you will have effectively 28 “extra HP” — melee attackers will suffer on your hands. The beauty of this combo is the synergistic affordability of both elements: once you cast AoA again, you will refresh your THP + your Arcane Ward, so with a little math dedication and some good resource management, you will be a tremendous melee tank punishing hard those who dare to touch you.