a person who tells everyone who he(she) likes / a person who circulates rumours about who he(she) likes
I don't like the kind of person who tells everyone who he(she) likes just so that he(she) can have everyone help
「言いふらす人」= spread rumours/circulate rumours/tell everyone ~
「自分で自分の好きな人を言いふらす人」= a person who tells everyone who he(she) likes / a person who circulates rumours about who he(she) likes
「みんなに協力してもらいたいからって、自分の好きな相手を自分で言いふらす人って好きじゃないな。」= I don't like the kind of person who tells everyone who he(she) likes just so that he(she) can have everyone help