When playing Lyra as a CP Carry, you should focus on getting in damage while healing (if possible). Start Lyra in the lane with Crystal Bit and Swift Shooter. As every carry/laner should do, try not to miss last hits on minions, and a really helpful trick is to poke the enemy carry with her basic attack (range 6.6) and her Imperial Sigil to lower the enemy's health for better positioning. Stand right next to the minions to guarantee every last hit, and it'll get easier when you get the Stormguard Banner or the Stormcrown.
If you want to be good at her, it's all about timing, from placing your Imperial Sigil, canceling enemy's abilities with Bright Bulwark, to slowing the enemy carry's attack speed with Atlas Pauldron + Arcane Passage combo.