The OP asks " ... how should I go about doing this."

Your property appears to be rather flat. Coupled with close neighbors, things can get sticky.

Start by getting someone other than yourself to document the existing grades over all of the project area. Use an engineer, surveyor, LA, or one who has many years of experience. The result should be a contour map with spot elevations also shown. The most important thing is the location where water is crossing the tract boundary.

Before starting the design, study the drainage law of your area. Law differs from state to state, but the most common applicaton is the civil law doctrine of drainage.

Here I repeat what I said on another thread. Note that part in bold.

A good overview can be found at:
http://publicworks.co.marion.or.us/engineering/DevEng/drainagelaw.asp

"The lower owner must accept water which naturally comes to his land from above, ..."

In Arizona:
https://www.coconino.az.gov/uploadedFiles/Public_Works/Homeowner_Guide_To_Drainage.pdf

"A second drainage law is "The Civil Law Rule of Natural rainage". This rule states that a landowner has the burden of receiving onto his/her land the surface water naturally falling onto and freely flowing onto it from land above it. The rule further states that a landowner has the right to have the surface water falling upon his/her land and naturally flowing onto it to flow freely from it onto lower parcels as it would flow under natural conditions. The key words are "naturally flowing". Any modifications to the surface topography or the drainage ways that cause natural flow to enter or leave a property differently (different peak flow elevations, frequencies and flow locations) can create a liability and put you at risk regarding litigation."

Or from:
http://web.aces.uiuc.edu/vista/pdf_pubs/DRAINAGE98.PDF

"A landowner has no right to obstruct the flow of surface
water. The owner of lower land obviously has no right to
build a dam, levee, or other artificial structure that will
interfere with the drainage of higher land, according to the
CIVIL LAW as it is applied in Illinois."

Really nice flat land. We have similar issues with heavy rains.

From your description it appears that the neighbors and your water are draining into his french drain. You should investigate building a dry-well of your own or of building a rain garden instead of altering your topography.

Actually, the two yards are not as flat as they look. Maybe they look flatter as I am photographing from the roof. There is a nice slope maybe a two foot difference from the house level to that corner for both properties. Although, it does flatten out a bit in that corner.Water never really "overflows" the wet area. The rest of the yard stays very dry.

I hear what you're saying about getting things checked out. Don't want to get the neighbor mad. Here is another option I thought of. What if after I put up my new fence, I built a deck about a foot off the ground covering that entire wet area. This way, the water can still flow and the ground can stay mushy, but I don't have to get back there with a lawn mower or have to worry about anyone walking around in the mud and tracking it in the house?

My previous place had similar problems. While I did not completely "solve" the problem I did make things much better by taking steps to improve the drainage of my soil. I had mature trees to deal with so digging big holes and rotor tilling were not really options. I did see very noticeable improvements when I switched to organic lawn care methods, spread compost a couple times and applied Nitron A-35 twice a year. As mentioned by gulfcoast8888, rain gardens are a solution that has been discussed on this forum.

BTW, I was also told that even though my wet area was below the street level that I could have somebody install a pump system that would pump excess water to the street. Personally, I did not like the idea of adding to the water run off issue.

- Brent