So I'll give you two sets of instructions, one for 32-bit and the other for 64-bit Linux Mint.
64-Bit Linux Mint
Download and install getlibs-all.deb. Open terminal with CTRL-ALT-T and then type these two commands:
sudo apt-get install lib32asound2 lib32gcc1 lib32ncurses5 lib32stdc++6 lib32z1 libc6 libc6-i386 lib32nss-mdns
sudo getlibs -l libnss3.so.1d libnssutil3.so.1d libsmime3.so.1d libssl3.so.1d libnspr4.so.0d libplc4.so.0d \ libplds4.so.0d libgnome-keyring.so libgnome-keyring.so.0 libgnome-keyring.so.0.1.1
Once you've done these steps, follow the 32-Bit Linux Mint instructions to finish installing TweetDeck.
32-Bit Linux Mint
Start by downloading the Adobe Air .bin file from this link and Tweetdeck from here.
Now open terminal with CTRL-ALT-T and navigate to your download folder. In terminal type:
chmod +x AdobeAIRInstaller.bin
Once that is done installing return to your Downloads folder in Nautilus. Double-click on the TweetDeck_0_37.5.air file located in your download folder and follow the prompts and the 32-bit version will be installed in Menu -> Accessories.
EDIT: Fixed some misspelled package names and instructions.