Below are the challenges you need to complete to obtain the ten new weapons in the Karkand expansion.
AssaultFAMAS10 Revives
10 Heals
L85A2 (unlock FAMAS to enable this challenge)
100 Assault Rifle Kills
20 kills with Underslung M320. (Can be the separate gun or the attachment)
Win 5 matches of Squad Deathmatch
EngineerG5310 Repairs
1 Infantry Repair Torch Kill
QBZ-95B (Unlock G53 to enable this challenge)
50 AT RPG Kills
1 Enemy Vehicle Destroyed with Repair Tool (Great tip from BadTriangle, destroying enemy equipment counts for this)
5 Conquest Wins
SupportQBB-9520 LMG Kills
2 Mortar Kills (Beware, you must rank up four times - 40k exp in Support to unlock the Mortar)MG36 (unlock QBB-95 to enable this challenge)
100 LMG Kills
50 Suppresion Assists
50 Resupplies (When going for the LMG kills just throw ammo packs for snipers on your team)
ReconQBU-8820 Sniper Rifle Kills
5 Laser Designations (multiple ways of doing this, SOFLAM/CITV seat in a Tank/Laser Painter in Attack Helicopter)
L96 (Unlock QBU-88 to enable this challenge)
50 Headshots
50 Spot Assists (BACK button,)
5 Knifes (Hold RB)
GeneralPP-19Arm 10 M-Coms (Rush)
10 Flag Captures (Conquest)
2 hours played on Strike at Karkand
MK3A1(Unlock PP-19 to enable this challenge)
10 Kills with PP-19
5 Kills in DPV Jeep (Transport Vehicle)
10 Kills in BTR-90 (Infantry Fighting Vehicle) (Best found at the RU deployment or Construction Site on Gulf of Oman)
2 hours played on Sharqi Peninsula
2 hours played on Gulf of Oman
Once you've unlocked all of the weapons remember you need to get 10 kills with them

Edits: As mentioned in the comments section by Cenrail, you don't have to actually unlock each weapon, you can just pick it up from a someone else who dropped it and get the 10 kills needed.
Also matdan12 mentions, these requirements don't have to be met on the new maps. Any maps will do.
As Scruffy McGuffy, DBB Klikopedia and BigWorthless confirm, none of the challenges have to be done in specific classes apart from the obvious ones, snipe kills, LMG kills etc Therefore it's best to unlock the complete first tier of challenges first. Try to get 2 hours on Strike at Karkand done so the time starts counting for Sharqi and Oman as soon as possible too.
If you get the 10 kills with every weapon and it doesn't unlock. Simply get 10 more kills with the Famas in one match and it should unlock (Any DLC gun may work but the Famas definitely does)Not sure why this happens, but it seems to be common.
If that still doesn't work then you may need to get 10 kils with each weapon in one sitting, It seems that some counts are being reset when turning your 360 offThanks for all the extra tips in the comments, keep them coming and I'll keep adding them in.