Discovery
You will get these 2 by entering the 2 new levels of COLDTV - Antarctica and the abandoned stations - for the first time. Nothing much to say to it, while playing the DLC you will get them eventually.

Lose to [colour] x amount of times.
These are all pretty self-explanatory. While you will play, you will probably get caught off-guard by these colours more than once, and when you have been playing this game for a while, you'll probably get the Cyan one the moment you start up the game.

(I know I did, and I even lost my save just before the DLC showed up.)

Send Cyan to space on Saturn.

You should get this achievement immediately since you are already in space
For this achievement you have to use the Endless Mode "Saturn" (accessible by pressing the spacebar on the main menu and paying the 300 CP fee).
While playing, keep one hand on the spacebar.
With keyboard controls you can also press space in case the colour and word match up. Do this 10 times on Cyan and you will get the achievement.

Supercalifragilisticexpialidocious.
For this one just start up a game of OLDTV and type out "Supercalifragilisticexpialidocious". When starting you have about 15 Frequency, which is a decent chunk of time, so you can take your time and type the word out slowly if you are prone to making mistakes. Once typed in, the achievement will immediately pop.
(And no, it doesn't work in the main menu.)

Pass Colour Roulette before it changes.
You will get this one probably before you even notice you were working for it. In colour roulette the word changes colour every now and then, but if you are fast, you'll likely just switch to the next channel without realizing that this was a colour roulette challenge.

Type Trump on Orange.
When the word Orange in Orange colour appears, you can instead of left-clicking build a wall and make Mexico pay for it type "Trump" and the channel will pass. Do this 10 times to get the achievement. Just be wary when playing so you don't accidentally just left-click. After all, you need to grab those achievement-getting opportunites by the ♥♥♥♥♥, when they appear.

50 CP with one use of double CP
COLDTV changed how "Double CP" works. Instead of being a one-time usable for the next run like "2nd Chance", it's now active for a minute and you can even pause it by clicking on it again in the skills menu. Once you bought the skill, immediately start the game since it doesn't wait and will start trickling down in the main menu. The easiest way to get this is to use Sherlock Mode, but if you aren't fast enough to consistently trigger it, you'll have to keep trying. The won CP aren't lost when you accidentally fail, all you have to do is essentially get 50 CP in 1 minute.

Completion.
You will get this achievement when you beat the abandoned stations. Pretty self-explanatory, but since the abandoned stations are not an easy task, it could take a while for you to get there.

Complete the abandoned stations with 60+ Frequency.
This achievement is pretty much only possible if you can trigger Sherlock Mode consistently. I would suggest starting in Antarctica to give you a sufficient headstart to finish the abandoned stations, since not only will you start out with more CP, but you will also have a full combo already, which makes triggering more Sherlock Modes in the abandoned stations an easier task.

Pass with only one letter visible
In the new levels, one of the new obstacles is that the word will only gradually appear. This is obviously very hard to do, not only because it's hard to guess the correct word based on only one letter, especially if it's a letter that multiple words can have. Not only that, but the next letter will appear very fast after that, so it essentially will come down to luck for many of you.

Pass blurred image without pressing space
Name and title kind of give away what you have to do. The word will be very blurred and tiny until you press space. You have to pick the correct choice without pressing space in this one. Keep in mind that this obstacle only appears in the abandoned stations (and Endless Mode if you found it once).

Answer a question about every colour
This is another new obstacle in COLDTV, where instead of a coloured word it will give you a question, for example what the colour or word of the last channel was. You have to type the answer for this one on the keyboard. This isn't hard in of itself, but it only appears on Antarctica (rarely) and in the abandoned stations, which are pretty difficult due to the other new obstacles. You also have to get lucky and get a question for every colour, which depending on your luck, could take a while.

According to observations by some users, playing on a language other than English might cause the achievement to not unlock. I would suggest turning the language momentarily to english if you are having trouble getting this achievement.

Thanks to CycoKillah and others who found out about this.

Survived the arctic mechanics
For this achievement, you have to beat all of the new mechanics in the antarctica, which are:
Colour Roulette
Gradually appearing letters
Blurred image
Question

If you successfully passed each one of these, you should get the achievement.

This should explain anything you'd possibly have questions about for the achievements of COLDTV.
If there is anything left you'd like to know or you think I'm wrong about something, do tell me in the comments, insults are greatly appreciated.

Also, once you're done, maybe it's time to stop playing stupid videogames.