TV online depends on your channel provider. Cox is a channel provider and Roku is another channel provider. A channel provider collects various television channels and distributes to its subscribers over a particular medium. Mediums include radio waves, coaxial and Internet. It gets confusing when a provider distributes through multiple mediums. It also gets confusing when you subscribe to HBO through Cox but can access HBO GO through Roku.
Roku collects its own television channels and distributes over the Internet, such as PBS, The History Channel and A&E. Roku either won't or can't broadcast the programming of other providers, such as Cox, Verizon and Comcast. Therefore, you can't watch Cox through Roku.
If you want to watch a long lost show that Cox no longer offers either online or on-demand; you need to subscribe to another provider, such as Netflix, Vudu, iTunes or Hulu Plus. After subscribing to another channel provider, Roku can deliver it to your TV or, if your TV has a built-in browser, you could add the particular app to your TV browser.
However, if you found a program on the Cox website that's not available through Cox on-demand, you can stream that program to your laptop and then transmit that program onward to your TV. However, there is a caveat: you have to subscribe to at least the Cox TV Essential package. If you do subscribe to a Cox TV package, you transmit that program from your laptop to your TV either wired or wirelessly.
Log into your Cox account and navigate to My Connection-->Watch-->Watch TV Online. With a wire, you transmit that program to your TV with a VGA, DVI or HDMI cable. However, using a cable means your computer must be relatively close to the TV to make the connection.
To transmit wirelessly, you have to research what's available on your laptop. Does your laptop offer any type of built-in wireless video chipset? This doesn't mean a wireless network card. A wireless network is different technology. I'm describing a wireless technology to transmit audio/video from your laptop to your TV.
So, as far as hardware requirements, you'll have to research what's available on your laptop. If you have a newer laptop, it may contain a wireless video chipset to stream audio/video to a compatible TV...or a wireless receiver-adapter connected to your TV. There are a few protocols for this technology, such as Intel WiDi, Wireless HDI, Wireless HD, etc. Research the documentation of your laptop or do a keyword search for "DLNA"..."HDI"..."WHDI"..."WIDI"...or "Wireless HD." If your laptop contains any of these chipsets, it will list the hardware requirements.
If your laptop doesn't offer a wireless video chipset, you could purchase a wireless USB PC-to-TV transmitter (RF-Link, Veebeam and Diamond Multimedia).
To transmit from a smartphone or tablet, you can research Cox TV Connect at http://forums.cox.com/forum_home/tv_forum/f/4/t/4373.aspx.