How can I go about Sound proofing my apartment?

Since their tv is inconveniently located on a common wall with your apartment, the only thing I can think of is this: purchase some peel-and-stick carpet tiles and apply them to the wall.. they are easy to remove, and shouldnt damage an apartment wall, and may serve to deaden the sound a bit

Living in a multi-family dwelling, like an apartment building, can often be noisier compared to a single-family home in the suburbs. People living in an urban setting generally accept a louder environment – whether that noise is from thin walls, old windows, construction, or street noise. You may not be able to completely remove the noise but there are usually ways to diminish the amount or reduce the decibel level. There are several apps on the market that can measure the decibel level in your home. If the reading is high enough, it might be something to bring up with your landlord, especially if this noise is controllable.

There are plenty of sites that offer sound reduction materials for residential spaces but if they are too expensive you can use more DIY methods. Decorative fabric can cover some of these industrial materials or you can devise a way to hide the material behind objects. When it comes to soundproofing your space, there are generally two types of materials: sound blockers and sound absorbers. To keep noise from entering your space, look for sound blockers. These products can be applied over doors, windows, and walls or in between cracks where these items meet.

If your walls feel “thin” and you are able to hear your neighbors TV through your walls, you can do a few things. Adding bookshelves and other thick pieces of furniture can help. You can make it even more noise-proof by adding a thick piece of foam between the wall and the piece of furniture. You can aslo cover them with hanging fabric.

http://homeguides.sfgate.com/soundproof-wall-cheaply-43537.html

There isn't any simple, quick, or inexpensive fixes to this problem. Older apartment building often did not do anything to sound-proof between units. The simplest thing that you can do is to add fabric to the walls and to move large pieces of furniture to dampen the sound.

As far as your landlord's attitude about this, almost all municipalities have noise ordinances and you may want to consult with your state or city's tenant's rights office. You mention that it wakes you up. If that is happening during the day, there probably isn't much you can do about it. But if it's happening at night, or early morning, you could ask the police to have a talk with your neighbor. Where I live, excessive noise after 10 pm or before 7am (9 am on weekends) is called "disturbing the peace."

Either take legal action, which still may not help, or move.

Caulking isn't going to make much of a difference unless there are open gaps the sound waves are traveling through.

Sounds like it's time to up the game. Call the police every time you have an issue for a couple of weeks. Document when you called and if the police come. After there is a pattern of disregarding the police, get an appointment with the chief's assistant and present your case. Try to make them do more than "just a warning." Also contact the housing authority. Most places, the housing authority is very serious about tenants' rights and will apply pressure to your landlord. If you can get some of your neighbors to also complain, that adds even more leverage. The health department or animal control can help with the dog issue.

Most importantly, DOCUMENT EVERYTHING. That way you have legal recourse if your landlord doesn't do anything about the problem or decides it's easier to give you grief than your neighbor.