You earn clan XP by playing with at least one other clan member. Playing solo does not net you any rewards.
Placing 1st, 2nd, or 3rd in a clan war (can be done solo) will also net your clan a boost in Clan XP.
Playing in Clan Vs Clan you can earn XP for your clan as well but I think you still need to be teamed up with at least 1 other clan member.
The page listed gives more detailed information: Call of Duty® Ghosts Clan XP & Rewards FAQ
In order to earn CXP, you need at least two or more clan members to participate in earning CXP.
I'm waiting for an update so that it can be possible to earn CXP while playing solo, it can be possible but through different game modes.
Example: Gun Game, One In The Chamber(If Released) FFA,
But until then, you will have to play with clan members to earn CXP.
Play any Online Multi-player Mode with At Least (1) Clan Member up to (9):
- 50 Clan XP per Clan Member per Win: 450 Clan XP MAX
- 10 Clan XP per Clan Member per Loss: 60 Clan XP MAX
Participate in Clan vs. Clan Playlists that requires (4) Clan Members:
- 100 Clan XP per Clan Member per Win: 400 Clan XP TOTAL
- 20 Clan XP per Clan Member per Loss: 80 Clan XP TOTAL