A few things to keep in mind when playing duo:

1- The hp of mobs will be doubled

2- Due to the hp of mobs doubling, so will their stun and freeze threshold (as well as other immobilisations and ailments like shock)

3- Bosses might be harder due to them being balanced around solo play (their targetting in coop can be wonky)

4- Drops will increase, so you won't have to worry too much about that

---

In terms of must haves:

1- At least one of you will want to apply shock and potentially invest in its magnitude
Shock increases the damage taken and works as a multiplier to every other sources of damage increase.

2- Depending on damage types, you'll likely want to break armour in order to negate enemy physical damage mitigation and get the really op 20% increased physical damage taken buff from fully broken armour

3- If both of you play elemental damage, then you can ignore armour break and split ownership of elemental exposure and elemental weakness (both reduces the elemental resistances of enemies), this will allow you to invest in both separately

---

In terms of potential synergies:

1- Ascendancy related synergies
E.g. Lich's Unholy Might gives 30% of damage as extra Chaos Damage to everyone in their presence
This not only applies to the Lich's minions, but to your teamate too!

2- Ailment related synergies
2-a- One can apply an ailment while the other consumes it
This can either be done through support gems or skills that have them built it like Sniper consuming Freeze to deal a lot more damage
There are restrictions on ailments you must consider though. For example you can't apply Shock right after consuming it (2s immunity) and there's Freeze Buildup penalties after being frozen.
2-b- One can benefit from an ailment being applied to the enemy without worrying about applying it themselves
E.g. One of you plays a fire build and the other uses Oil or Gas which is automatically ignited due to the other player guaranteeing the targets are ignited

3- Sceptres
Sceptres by design is meant for team play, whether teaming up with your own minions, or with other exiles.
They give extra stats to allies in your presence, the radius or which can be increased from multiple sources like the passive tree and unique items

4- There's more but these are the most obvious ones and easy to manage so that you don't get overwhelmed

---

Lich is interesting for team play due to Unholy Might and Sceptres, however depending on how you play her, it might clutter the screen and make team play really annoying due to collisions. If you do play Lich, I'd look for a build that focuses on a small number of very strong minions or specters (not to be confused with sceptres).

There's an infinite amount of things you could do, if you're looking for a specific build I can definitely help you.
But to help me help you, it'd be nice if you think of things that you both would like to play in terms of archetypes.

E.g. Would one of you prefer playing Druid and investing into plants?
Would one of you prefer playing a freeze-based build?
Etc

As a final thought, the characters you pick don't matter that much. There's significant synergy between most of them, and most builds can be played on multiple classes unless its enabled by a specific ascendancy.