It will depend to a degree on what the instruments are, and how advanced the player is.On piano/keyboard, it's quite easy to play both parts.On guitar, which is,I guess, what you play, it's not easy.Especially in C. You may well be able to transpose the key to E, using E,A and B(7). Or A, using A,D and E.That way, at least, you can put a bass pattern in on open strings underneath a blues tune using the top 2,3 or 4 strings.
If the player is someone like Martin Taylor, then one instrument is all that is needed - he'll put in bass, chords and tune at the same time. Us mere mortals tend to use another person to play the 'missing' parts.
The pattern you quote sounds better if the last F and G are swapped over, and usually the last chord is a G, used as a turnaround to get into the next verse.Leave it as C the last time though !
The pentatonic you quote isn't clear (to me), and is (in C) C,D,E,G,A for the major, and C,Eb,F,G,Bb for the minor. To make your tunes sound more bluesy, play Gb as well.