Tip overview

Get Nimble from the item shop. Being lighter on your feet and with proper positioning (especially with warder's field enabled), you can circle him counter-clockwise (away from his gun). This allow you to hit him without him ever hitting you.

Remember to use your ultimate after his is on cooldown, otherwise he will not die, not yet amigo.

Get the treacherous ground legendary. Then just hit maeve.

Be extremely careful with your positioning! Grover has very good anti-sniper capabilities. You don't want him to notice you first. Remember about stealth, move and look around often to avoid a nasty suprise which is an axe hurled from across the map.

His projectiles have a huge bullet drop, flying a curved line, and do blast damage. Keep that in mind when dodging.