Don't buy packs. Buy specific cards that buff splinters you like to use. Level up a summoner that you like, and then buy cards that fit that archetype.
Got to around 1200-1300 in bronze doing that and am now 1900ish in silver after about a week or two of play (and about £100 invested, you will need to spend to be able to compete unfortunately).
HI zedcell, Thank you for the advice!!
I'll buy some cards for now to reach bronze I, and then enjoy the game for a while before spending more cash.