As a proud *ethnic* German, I want to ask if the German Empire or any state that is similar (lets say a Germany formed by the dying HRE. Also I wouldn't like to Austria count as "Germany") colonised larger portions of terriotry and fully assilimated a population with settlers (like USA, Canada and Australia today), and at least have twice as much land than OTL.
Bonus if its the actual German Empire (1850s-1918).
Extra bonus if they seize it from Britain or France!
Why no Austrians?
You want a HRE Germany formed without Austria - one of the largest constituant members of the HRE?
Add Austria and you remove a lot of early tension between a spurned Austria and the, presumably, prussian dominated Germany, and massively increase the new nations power. Also more balance with North and South, Catholic and Protestant would also remove a lot of early problems.
'Germanising' Poland, Bohemia, Croatia and any other lands makes Germany a huge contiguous entity. Which would not have the likelihood of losing all that territory to independence movements.
To have the overseas empire you seem to want, with the conditions you imposed, you need really early unification so that there is still a number of viable colonies available, no Bismark as he seemed generally anti these kinds of adventures, masses and masses of spare money and population and most importantly a Kaiser who is keen on the idea and doesn't mind annoying all of his neighbours.
Overseas colonies for a land based power like Germany- not a huge coastline or naval tradition there- require masses of investment not only in the colonies themselves, but in a navy too. Such naval buildup leads to trouble with already established naval powers. So you need either a fantastic diplomat or a skilled admiralty. Or ideally both. The first to keep the peace while you expand rather aggressively and the second to win a naval war against all odds should it come to that.
Now you just need to keep the colonies happy, or see them drain even more resources. Whether in infrastructure or independence movements.