There's more to it than simply having a wireless router. The wireless router just creates a network of wired and wireless devices. It's sort of like asking, can I use the roads in my neighborhood to build a house. You certainly need the roads to deliver the construction materials, get the workers to the job site, etc., but building a house involves a lot more than simply having roads available.
First you need a source, some place where the content (movies) is stored. And you need to make those sources available through some sort of protocol (let’s assume Windows file sharing for now). Then you need a media player on the receiving side to pull that content to the TV for playback. If your TV is DLNA enabled, then it will already have an ethernet port and media player built-in (wireless may be an option and require a separate purchase for the adapter). And it’s capable of only playing certain file formats. As long as those file formats are compatible w/ your available sources, you’re in business. You just make your wired/wireless connection for the source and TV, configure the TV to pull content from the shared files, and enjoy.
If the TV is not DLNA enabled or otherwise doesn’t provide an ethernet port, then you’ll need a standalone media player to place near the TV (e.g., WD TV Live). In this case, the media player itself has all the connectivity, pulls the content, has to be compatible w/ your content sources, etc. Then you connect the media player to one of your TV’s available inputs (composite, component, HDMI, etc.). IOW, your TV is just a simple monitor, it displays whatever the media player shoves its way, but otherwise has no idea how that content was obtained.