I would recommend attempting this on your normal run as there is no achievement for beating easy and normal is easy enough to give ample time for completion. A good normal speedrun will take approximately three hours. If you attempt this after your hard run, you will gain prior knowledge on boss patterns, enemies, and which way to go.
Here is a few tips :

1. Jump/Avoid as many enemies as you can. When moving from waypoint to waypoint, jump all the enemies in a row.
2. Make sure you collect whatever upgrades you are close to. Check your map often, as most upgrades only require a detour of a minute or two, and in the grand scheme of things make the boss fights go far quicker.
3. Once obtained, make use of your plasmas. If you do need to kill enemies, explosives take them down fastest. Also spam Eagle power for the sections where you are locked in until the enemies are defeated.
4. The cold plasma is great when speed running, especially when used with the Kunai attack. You can freeze enemies and turrets as you run past them to stop you taking damage.