The institutional trading business is customer driven. As a trader I provided liquidity and execution for these large mutual funds, pension funds and hedge funds. Along with the customer side we also trade proprietarily for the house. The plain vanilla execution business works something like this. Customer A wants to sell 1mm shares of GE so I would piece the order to the floor to get the customer the best execution. That's why you might see a repeat seller in a name. The providing of liquidity is where things get interesting. Example; Customer B might come in saying bid wanted 1mm GE and it is my job to price the 1mm. GE is trading at 36.5 to 36.62 and I bid 36.35 for 1mm. The customer hits me and I own them. Remember this is negative selection, meaning I did not want to own GE and now must trade the position. (we do get paid a commission but if a trade goes against you it doesn't matter). My average volume for a day was about 4mm shares. Throughout the day I traded both intraday and longer term prop..
Career Path: I graduated from a second tier school with a 3.8 GPA and couldn't get a job on a BB trading desk so I took a job in the back office and worked my ass off.(luck didn't hurt) That led me to work on the NYSE floor for a little while and I learned a tremendous amount. Shortly after that experience I landed on the trading desk.
Cleanup Prints: I am only speaking for listed stocks. Not all large prints are cleanups. Many large institutional orders get started with a print. For example a customer may want to sell 3mm GE and say to me trade 1mm to work 3mm. So I px 1mm, trade it and begin to work with the customer selling the 3mm. The same may be done on the back end of an order but how can you tell the difference.
I went over this in pretty broad strokes. I hope I answered some of your questions. If I didn't let me know I'll take another stab at it.