Update from Snow Leopard to El Capitan fails on Macbook pro (Mid 2010)
The installation fails due to a crash caused by the SystemMigrationUtils process while it tries to talk to the battery of your MacBook Pro:

Jan 27 11:29:21 […] Thread 8 Crashed:: Dispatch queue: com.apple.root.default-qos
Jan 27 11:29:21 […] […] 0x000000011b3adff7 logBatteryState + 963


The fact that this fails is indicative of an issue with the battery of your MacBook Pro.

To fix this, first be sure that you are not only connected to the mains but also have a working battery actually attached to the logic board of your MacBook Pro. (Source: Apple Communities)

Second, reset your NVRAM by turning off your Mac, holding ⌥⌘PR and turning your Mac on again. Make sure to hold the buttons until you hear the startup chime for the second time.

Shut down your Mac one more time, then reset its SMC by pressing ⇧Ctrl⌥ and the Power button at the same time. Your Mac needs to be connected to a power supply during that entire process. Keep an eye on the little light on the MagSafe plug while you hold down the for keys. Release the keys at the same time. The MagSafe light should change colors for a second or so; if it doesn’t, retry until it does. Now try installing El Capitan again. (Source: Reset SMC of a MacBook Air, MacBook Pro Retina, or MacBook Pro’s with an internal non-removable battery)

If the installer still crashes during logBatteryState, try resetting the system management controller by physically removing the battery from the logic board. To do this, you’ll need to remove the screws from the bottom case, lift the case just like you did back when you installed the SSD. Locate the battery, then remove the three screws to detach it from the logic board (two screws are in plain sight and the third screw is hidden under a warning label.)

Now lift the battery a bit, remove the connector from the logic board, press the Power button and hold it down for 10 seconds with the battery still disconnected. Now reattach the battery and retry the El Capitan installation one more time. (Source: this comment on ifixit.com)

If you have followed all the steps and the crash still occurs during logBatteryState, you might want to get another battery of the same model that is confirmed to work, then install it in order to figure out whether there is an issue with your battery; if not, it might be an issue with the PMC on your logic board.