Hi, Peru has many places to birdwatching, but we are in the rainy season now.
In the North, from La Florida are beautiful cloud forests, here you can try to see the Marvelous Spatuletail, then you can visit Abra Patricia (Royal Sunangel, Ochre-Fronted Antpitta, Bar-Winged Wood-Wren, Johnson’s Tody-Tyrant, White-Capped Tanager) this area is wonderful for many tanagers, hummingbirds and several endemic birds, down to Moyobamba is excellent to lowland birds, then you can take a flight to Lima from Tarapoto (2hours from Moyobamba) if you have time you can continue to Leymebamba (Endemic Buff-bellied Tanager, Gray-breasted Mountain-Toucan) then Cajamarca.
In the south, Cusco, you can visit Abra Malaga, excellent place for birdwatching in Andean Paramo, Polylepis Forest (Royal Cinclodes) and cloud forest.
The famous Manu Cloud Forest, (White-collared Jay, Mountain Cacique, Grass-green - & Golden-collared Tanagers, Hooded - & Scarlet-bellied Mountain-tanagers, Quetzals, Masked Flowerpiercer) in this place I recommend no less than 5 days, one of the best places in Peru.
Another great place is the Sanctuary of Ampay, a forest of Podocarpus in the city of Abancay to 4 hours from Cusco, great place to see endemic birds (Chestnut-Breasted Mountain Finch, Apurimac Brush Finch, hummingbirds and the best place to see Undulated Antpitta The last time I saw 4 and heard at least 20.
Machu Picchu is also an excellent place for bird watching, if you travel to see the citadel, a couple of days would be nice.
I've been several times in these areas during the rainy season, you can not expect good weather every day but with a little luck can make excellent check lists.
There are also good places in central Peru as Unchog Forest, Carpish Pass and the famous Paty Trail.
If you have questions, I will be happy to help you
Grettings
Steve