Could I install my OSX system into and boot from the external hard drive I have a model of Retina MBP late 2013, my internal hard drive is failing and I do not want to by an ssd replacement...(quite expensive!).
I have a Sumsung MZ-750250B with a hard drive enclosure and with usb3.0 connection cable. Now could I install the OSX system into this hard drive and boot from it every time? And whether the reading and writing is fast enough?

Thank you.
Yes, this is possible. For full details refer to How to set up and use an external Mac startup disk. If you have any problems, feel free to come back here and search for an answer (or ask another question if you can't find a solution).

In terms of your Samsung MZ-750250B, if you actually meant a Samsung MZ-750250GB then this is actually a SSD and supports a sequential read speed of up to 540 MB/s and a sequential write speed of up to 520 MB/s. I think you'll find this is more than enough for everyday use.