How to view video/photo from smartphone to TV? Is it possible to view media contents from the smartphone to the TV?

I would like to buy a movie from the PlayStore but I am not sure on how to view it.

I need a specific cable (to connect the HDMI port) or is possible to use chromecast too?
You didn't say what phone you have, or what Android version, so here's a summary.
Physical cables
If your phone supports MHL
and your TV does as well (almost all HDMI TVs do), then you can connect the phone to the TV with an MHL cable. There are two kinds of cables: a standard one, and a special one for certain kinds of Samsung phones. See How does MHL work? for more details. Usually, the TV mirrors the phone screen (i.e. it displays the same thing at the same resolution), but Android offers apps a way to display different content on the TV screen. This is commonly used by media players and presentation apps.
The oldest HDMI TVs don't support MHL, so you'll need a slightly different type of adaptor to connect your phone. You can tell these adaptors because they have an extra input for an external power supply (often a micro-USB socket). Apart from that, it works the same way.
If your phone supports SlimPort
Some phones use SlimPort instead of MHL, such as the Nexus 4. You can't tell by looking at the phone: both SlimPort and MHL use the USB-on-the-go connector. You have to know which one your phone uses. With the appropriate adaptor, you can connect your SlimPort phone to any HDMI, DisplayPort, or VGA input. Just like MHL, once you've done this the TV screen will mirror the phone screen, or some apps can display different content.
If your phone supports HDMI
A few phones and tablets have mini-HDMI or micro-HDMI outputs, but this has largely been replaced by MHL and SlimPort. If you do have one of these outputs, you'll need the appropriate cable to connect it to an HDMI input, and just like MHL or SlimPort, it'll mirror the phone screen, or display different content in apps that support it.
Chromecast
If you have a Chromecast and Android 4.4 or later
and your TV has HDMI input, then you can use that to mirror the phone screen on the TV wirelessly. Plug the Chromecast into your TV, and use the Chromecast app to help set it up. As with MHL, Android also offers apps a way to display different content on the TV screen.
If you have a Chromecast and earlier Android versions
You can still use Chromecast with media player apps that support it, including YouTube, Play Movies, and the Gallery on some devices. You just can't mirror the phone display to the TV.