Brazil to Peru bus

3 weeks to make the journey, stopping off along the way? It's pretty rushed but certainly possible. Doing it in one trip would be nightmarish and involve multiple connections.

Most logical route would be through Bolivia. First take for a flight to Corumba, Brazil. Flights can be competitively priced to buses in Brazil, so worth it to save time. Cross the boarder into Bolivia and take the train or bus to Santa Cruz. From there you should consider visiting Sucre, Potosi, Salar de Uyuni (doing the 4 day tour starting in Tupiza if you have time, otherwise start in Uyuni), La Paz and Lake Titicaca (Copacabana, Isla del Sol). From Copacabana there are several onward connections to Cusco via Puno.

Online bus reservations are almost non existent in Bolivia. Best method is to either book at a travel agency for an undisclosed fee, or go to the bus station and book yourself in person. Try to get a look at your bus if possible as standards vary enormously. Bus cama (bigger seats) can be worth it for longer haul trips. Most importantly of all, bring plenty of warm clothes for overnight trips through the Bolivian altiplano. It will be well bellow freezing and the buses don't have heating.

Buses are generally pretty terrible in Bolivia. If your budget permits it's worth considering domestic flights which can be reasonably priced.

I agree with #1 the trip would probably be nightmarish. If you want to travel with a minimum of connections you can choose Ormeño. They run one weekly bus between Sao Paulo Cusco and Lima along the Transpacific Highway. The trip takes 96 hours.

Caracol has a weekly bus along the route through Bolivia. It is probably best to go to the bus station and book in person.

Rio to Lima on a bus? Kill me. Kill me now.

I agree with both posters, but would suggest the following itinerary if you have time to make some stops on the way

Sao Paulo - Campo Grande: o/n bus (Andorinha)
From Campo Grande some fazendas will pick you up for a Pantanal tour and drop you off on the highway at Buraco das Piranhas
Buraco das Piranhas - Corumbá
Cross the border (why do so many people spell it boarder on this forum...autocorrect?) to Quijarro
Quijarro - Santa Cruz: o/n train (very comfortable)
Santa Cruz - La Paz: fly. Definitely fly.
La Paz - Copacabana: short bus trip.
Have a couple of days on Lake Titicaca
Copacabana - Puno: bus
Puno - Cusco: bus or Puno - Juliaca by bus and Juliaca - Lima by plane

If you are tempted to get a direct bus, just try and imagine what it will smell like by the last day. I once joined a direct La Paz - Mendoza service on the last leg from San Juan to Mendoza. The stench when I bot on board almost made me hurl.

Oh, and to pick up from hazzdawg's comment on horrible Bolivian buses: avoid overnight buses in Bolivian vehicles or on Bolivian roads unless absolutely, absolutely necessary. I did Tupiza to Tarija and only did it thanks to benzodiazepines. TAM (the Bolivan military airline, not the massive Brazilian airline) and BoA do good, cheap flights.

Rio to Lima on a bus? Kill me. Kill me now.
Can I jump in. Kill me too!

We have about 3 weeks so time isn't an issue
I would say time is a massive issue. Considering distance and places to see along the way.

Sounds horrific. I would be tempted to follow on from Poly's idea. And use flights as much as possible.
I would look to go SP to Campo Grande by flight. And Santa Cruz to La Paz too. Also Cusco to Lima with a flight. That way you give yourself more time to actually see things on the way.
You may be able to find cheap flights to Santa Cruz (generally can be found) and then continue from there. You could go to Rurrenbaque or even to the Salar (be careful of Altitude).

Bolivian buses have copped a bad rap. I can see why - I saw some doozies. But I never really had a massive problem apart from the fact the seats are made for tiny bolivians and well, my frame (5'9-10ish") just didn't like that cramped space and constantly accidently putting my knees into the back of the person in front of me.