How can I farm teeth?

Relics are your best friend. Each relic you run increases the amount of teeth you get from completing a game. When I was running with decent people, I would generally run eight or nine. You can run pistols only very easily on point of contact (which is what I would recommend farming on) if you use the revolver, and never get the akimbo upgrade (demolishes your accuracy for negligible benefit at close range) or you can ditch everything but the grenade turret (all relics but the no machines relic), as the grenade turret is very OP on point of contact. As with all of these tactics though, you'll want to be playing with people that have some idea what they're doing. Playing with randoms may warrant taking off a couple of relics, depending on how skilled you are. Also there are several things you can do on a game to game basis to get more teeth, such as getting a completionist bonus by perfectly completing all challenges, and getting kills (as every few hundred kills or so you earn a tooth regardless of what point you are at in a game.)

If you own the DLC, I would say Mayday on solo is a pretty decent method of earning teeth. I usually pick up approx 15 teeth by doing ALL of the challenges. Mix this with the Point of Contact map (see other answer for more detail) to save yourself from boredom. I suppose it depends on how many teeth you need.

Downs
On solo, you can get downed 3 times by the time you get to the final area (you seem to earn extra downs, starting at 1, by progressing through the ship). I can get to the Kraken without a single down, although the final drill point can be close. Remember, this applies to solo only - not playing custom alone (i.e. there is a specific solo option in the menu).

Challenges
Aiming for all of the challenges is pretty easy. On solo I have never had a gun challenge (i.e. buy a gun and get x number of kills). Challenges are even easier if you bring flares with you.

Flares
You should be able to pick up 2 or 3 standard flares by opening boxes (remember that you can only hold 1 at a time). However, you can also craft sticky flares through schematics (each crafted sticky flare gives 2 flares, for which you can hold both at the same time). A blue print is found in the starting area (left side stairs if you are facing back out to sea).

Use flares/sticky flares at the start of the challenge if you feel it will help. Challenges like killing the marked aliens before all others are much easier if the aliens are stood still for a while. Also, killing aliens at a minimum distance.

Sticky flares also help a lot during the gas pipes section. Throw one down if you get into trouble (remember you only need to defend one, not all 4 pipes).

There is an exploit with regards to sticky flares. If you pick up an item that is in your left slot (e.g. normal flare, trophy or hypno knife) and you then craft a sticky flare, it will drop the normal flare/knife etc in the spot you are standing. You can then swap this item with your sticky flares. You can use this method/exploit to layout a set of sticky flares ready to use (because you can then carry on crafting the next item). The same works for venom grenades and items in the grenade slot on other maps.

Before travelling to the top deck make sure you have 2 sticky flares in your inventory and the sticky flare blue print (plus 3 red boxes if you can). This will mean you can use 4 sticky flares in the top area. Try to save 1 or 2 for the gas pipes section.

Venom-X
Make this schematic a priority. It is found in the main area before you go to the upper deck. The crafted gun will vary depending on the 3rd craft item you pick up. I prefer the blue and orange variations, but the green variation is also effective. I avoid the purple variation. As long as you don't pick up the 3rd craft item (i.e. open the red box but leave it on the ground) then the next red box should offer another random 3rd item. Use this knowledge to craft the version of the gun you want. However, you will lose red boxes for sticky flares.

Assuming you have a blue/orange/green version of the gun, use the gun to clear out rhinos and batches of aliens. Fire it into a group, or into their path (they take damage even seconds after you fire it into an area). Wait a few seconds before firing it again (if you need to fire it into the same area). Don't worry about running out of ammo, plenty of ammo drops (it is the glowing eggs dropped randomly by dead aliens, which gives you 2 shots).

The gun is much more useful in the upper deck, and in particular when you come to defending the gas pipes.

Drills
Other than all of the mandatory drill points, I also drill open the 2 doors in the first area (for red boxes) and the 2 doors in the main area (for the Venom-X and more red boxes).

Gas Pipes
I would say that this is the hardest part until you have done it a few times. Facing the pipes, you want to be standing on the single storey building or in front of it. Kill what you can, but don't worry about losing 2 or even 3 pipes.

A stowed riot shield helps the occasional alien behind you, as does a turret or drone (putting a turret on the bottom floor facing a pipe is effective). Kill aliens that attack from behind quickly. Use a flare if too many aliens appear. Use the venom-x every so often onto a pipe for easy kills (remember, it will also kill aliens that move in a few seconds later, too).

Kraken
The key is to actually concentrate on killing aliens rather than the Kraken. I often find cash can be more of a problem than the Kraken on this part. Immediately run to the right hand side into the far right corner. Standing on the red, hugging the turret, kill all the aliens that spawn. On solo, I only ever get 2 seeders, and they are pretty much straight away. After that, you get a mix of hunters, scouts and seekers.

Just keep killing, replacing ammo/armour with the cash you earn. Stowed shield is not really required because you are right up in the corner anyway. Switch guns when low on ammo so that your ammo drops are more effective (e.g. give more bullets for the cost). During intervals the aliens will stop spawning. Use that time to shoot the Kraken in the red dot. When it is glowing you deal a lot more damage.

Every so often the Kraken will place his tentacles on the ship, causing the ground to glow red. As long as you stay off the metal you will not take damage. Less aliens spawn here so get your shots in.

When the Kraken goes under water run across to the opposite side, and again on the red (off the metal). This opposite side seems easier to defend because aliens start spawning on the platform behind you but need to jump over to get to you. Pick them off.

Continue the pattern of clearing aliens, keeping ammo/armour up with the cash and shooting the glowing red dot when it is safe. I think you swap positions maybe 3 or 4 times before you kill the Kraken.

I never use the turret controls because I feel it is easier defending yourself by hiding behind the controls and shooting what gets close.

Class Setup
I don't think the class really matters, although I prefer the weapon specialist. For 15 teeth the class has an upgrade whereby you get unlimited ammo for a short time (without needing to reload too). This helps on the Kraken fight if you run low on cash. You also get fast reload (good for LMG's) and 50% extra bullet damage after all of the skill points are put into the class.

For ammo I prefer incendiary because it carries on causing damage after the bullet has landed. It isn't that expensive either, which helps you keep it stocked up.

I prefer taking the stowed shield and either a turret or drone (only really use this twice). I also take armour.

Skill Points
Evenly spread the points over ammo, armour until around 3+ for each. You can then use the pistol upgrades up to the point you can carry 2 primary weapons. Bringing an LMG, Venom-X, pistols and another gun to the upper decks is recommended for an easier run. I never spend points on the final pistol upgrade because dual wielding accuracy means you lose a lot of ammo.

By the time you reach the Kraken you should have max armour, ammo and class, and 3+ on pistols. The rest you can waste as you wish.

Relics
I have run my method through half a dozen times without any real problems, but I haven't tried with multiple relics. Having the relic that means you take more damage is pretty straight forward. It doesn't appear to do any extra damage until you lose your armour.

Achievements/Trophies
If you wish to obtain the achievements/trophies associated with this DLC it is feasible to do the entire DLC in solo. You could do the challenges and 1 relic achievement/trophy in the same run through.