The main reason you should not take smite in lane is because you cannot gain gold from minions if you have a jungle item, and smite is useless unless you have a jungle item. The solution? Don't buy a jungle item until after laning phase is over. This allows you to get some or all of your core items, and your jungler to max their own pet before you start farming the enemy jungle. The win conditions for this build are securing objectives, which is a garentee with two smites, and denying farm, both for your enemy laner and jungle.

Durring the early game focus on controling the wave and having more CS instead of fighting, you are down a summoner spell but have more wave control with smite, so playing passive is in your favor. If you see the enemy jungler gank see if you can either counter gank, clear the oposite side enemy jungle, or gank the opposite lane, depending on your possision, and the possision of enemies. Do not fous on plates, you get more value out of repetedly pushing minion waves and roaming than you do if you take turret, because keeping both tier one turrets up as long as possible makes denying CS to your laner much easier, and to make this build work you have to roam as much as possible.

Durring the mid, play arround the enemy jungle when objectives are not up, place deep wards along the way and kill the enemy jungler whenever you get the chance to. Return to mid lane whenever it is empty and the wave has pushed on to your side of the river. It is likely that at this stage of the game your ADC will get more value out of farming your wave than you would so let them. When objectives are up you can make the decision to either take the enemy jungle while they're destracted, push the opposite lane, or help on objective, though you should only help with objective if you believe that your team would not get it otherwise, if the objective happens to be drake soul, or if you know there is no opportunity in either of the other two option at the moment. If you decide to invade while objective is up, start on the opposite side jungle and path towards the objective, either the fight will sill be happening and you can finish the job, or one side will have taken the objective, in which case you can pick off anyone retreating, or if your team lost the teamfight, you can safely recall.

Late game is similar to the mid game with this build, except you can just start ignoring mid entirely. Keep farming the enemy jungle and push Mid/Top up to the turret and back off. Keep placing deep wards to help win the vision war. The main difference between mid and late game is that now you should be team fighting every time an objective is up, a barron or an elder drake could be the reason you win the game and with two smites and Lillias team fighting aptitude it is hard not to secure these late game objectives.