Buying more expensive cable is not the solution to your problem. Cat 5 cable support up to 100Mbps speeds, while cat 5e reliably supports 1000Mbps / gigabit speeds. Cat 6 and higher cables are needed when you want to have connections speeds that exceed the 1000Mbps threshold. Secondly, in order to attain speeds that exceed one gigabit, you would need to upgrade the ethernet devices on your network, i.e. network cards, switches, routers, cabling etc.
Installing ethernet cable in your house may require more effort compared to setting up a wireless mesh system, but it is the most reliable way of bringing high speed capacity as close as possible to your remote access points.