I have looked into it, but I can't find the answer.
When I say 'around', I mean walking along the perimeter of the lake and returning to the same place we started at.
Google Translate says that it is Burley Griffin湖を歩きます.
My guess is:
「バーリーグリフィン湖一周に歩きます。」
Is that correct?
So, let's talk about noun phrases that might be used to express the concept of a walk around something.
xx一周ウォーク
xx一周ウォーキング
徒歩でxx一周する
xxを歩いて一周する
The first two are most commonly used in terms of a predesigned course or trail that you might take around something. All these terms can be used with standard verbs such as する or 始める. However, there is a special verb that can be used to indicate completion of a walk 完歩{かんぽ}, specifically for the first two examples. It doesn't seem to be used in the case of the latter two presumably because you already said 歩. In fact, you can just say
xx一周完歩しました
to say you finished walking around something.
Here are some other ways you can say it, especially for a larger thing such as Lake Biwa or Japan (or I suppose if you want to make it seem like a long trip), you might say
xx一周(する)歩き(旅)
xx一周徒歩 (旅 or 旅行)
For going around the world (obviously you can't do it all by walking), it seems the more common way to say you completed it would be :
世界一周徒歩旅を達成した
