I've tried transplanting many times, sometimes i would use all sorts of techniques to transplant a plant to a larger medium. I will share some tips on how to do it successfully so you wont kill your plant doing it the wrong way (been there done that).
Good-to-know tip: When transplanting a plant into a larger medium, make sure the soil is nice and moist. Moist soil will stick to your finger, if the soil consistency is just right you can begin transplanting. Another tip, for clones and seedlings use a paper-cup, paper-cups are most easy to deal with with a transplanting situation... just cut the bottom of the pot and push gently so the soil mass goes up and releases from the sides of the cup.
In my situation im transplanting a plant from a 2 gallon plastic pot to a 5 gallon bucket. This can be tricky, usually transplanting a large plant such as this one would need 2 people to do this (one to hold plant securely and the other to pull the container off), but it can be done with one person. The trick is to tip the plant just slightly so it slides out, usually it takes a bit for it to budge... what i do is i use my index finger and i jab it into one of the drainage holes then push down, now this wont effect large plants but do so gently just enough to get it going... this works really well to get the soil detached from the sides.
You should have already prepared the bucket with soil. You can judge it and place the old container in to measure up how high you want the plant to be.... for me im leaving about 1ft.
And i want to leave about 4inches of no soil on the top.
Once the soil is the appropriate height you can place the plant ontop of the prefilled soil, proceed to fill the sides of the container with soil. Press down gently and continue filling the container each time. You need to use gloves (if your a wussy and don't want to get dirty then use a small shovel).
This never fails, you should be transplanting like a pro!