Well, you could do raised beds for things that don`t like wet feet or you could go with the flow and plant things that like moist/wet/boggy conditions.

Have you had a greater amount of rain lately or is this a low level in your yard so it is always wet?

Ideally the soil will not touch the bottoms of the fence boards as it will cause them to rot. So I'd recommend making the flower bed wider and sloping the soil up from the fence ensuring soil is not touching the boards. You could mound the bed up from the fence then back down to the lawn level. Two feet width is narrow for a flower/vegetable bed and yours looks to be less than that.

I would mix the native clay soil with the purchased garden soil. Clay soil can have nutrients that the purchased soil may not have and will also help with moisture retention. If you mix a few inches of the clay soil with the new soil it should give you a deep enough bed for most plants.

Any grass you dig out can be composted in a corner of your yard. You can pile it up with the chunks upside down and it will rot and become good soil.

I agree. Don't let the soil touch the fence because it will rot the fence.

I would suggest first widening the plant bed. Do a double digging method -- (dig out a hole, , dig a second hole next to the other hole, mix that soil with triple mix+ rough sand (brick sand)), then put that soil into the first hole, dig out a third hole, do the same mixing and place that soil into the second hole., dig a fourth hole, mix the soil and place ithe soil in the 3rd hole .. and so forth. Thie will aerate the soil, make it rich. The compost will break up the clay, the sand will give it some grit.

Any grass you dig out will make great compost. In my case I would just put the grass back, but inverted so it kills the grass. The decomposing grass will return nutrients to ground.

Then use an edger and dig a V-shape trench to define the flower bed edges.