Don't rototill.
Before condidering a lawn, consider addressing your drainage issues. Your lawn shold drain away from your house, so consider adding soil to create slope from the house. Then consider creating swales to drain the water further from the house and out of the lawn. Then possibly a french drain to carry the water from the swales to a collection area or the storm sewer system.
Once you have the drainage addressed, get a soil test so you know what you have (soil test will tell if you really have clay and what your soil needs to grow turf.) Try www.loganlabs.com.

Thanks for the response grass 1950. Where would you direct the water? Where to put the Swales? I feel like for all the square footage of my yard, there is nowhere to do anything!
Ive posted a pic of what my yard looks like. Flagstone at the house and under the pergola at the back, gardens all along the sides and its fully fenced.
Thanks!

I'm afraid drainage decissions should only be made by a person on site. It's the only way to make an acessment.
Look for the lowest point on your property (away from the house) and see if it's feasable to direct the water there. Also look to see if that area is close to a public street, alleyway, storm drainage ditch or storm sewer. If not, research the term "dry well". As you are aware, there is always the possibility of french drains.

This post was edited by grass1950 on Fri, Apr 19, 13 at 16:30

grass1950 is right - fix your drainage issues first before worrying about your lawn. I started fixing mine by hand last summer and plan to finish this summer (there are a lot of corrections needed here). You can do a lot of reading here on Gardenweb about drainage, swales, and grading.
Stand back and look at your property in relation to the surrounding homes, and see where it looks like the water was originally supposed to go. For instance, if I stand outside the front of my house, it's easy to see the slope on my street (from north west to south east), and the way the water flows over all. The grading on your property should nicely coincide with the overall grading in your immediate area.
It could be that landscaping by your neighbors has caused you to have more than your share of water, either because they have raised their grade or because they've diverted the original direction of the flow. Or perhaps previous owners of your house might have caused water to flow incorrectly towards your house with all their landscaping "enhancements".
I see from your first post you are in Ontario (as I am). If you cannot tell by looking at your house and the surrounding ones which way the water was initially meant to flow, you may be able to see the original plan that the city still has on file. (Or you may have access to old maps online or you may have received a map with your mortgage papers). That will tell you what the original design was and you can see how it's deviated from that since. But it may be easier to just go outside and look.
Even if the flow was oiriginally designed to come towards your house, chances are there was one or more swales that would have diverted the fllow to one or both sides of your house so it would have flowed out to the front and bypassed your house.
So do have a look as to what is going on. The solution may be as simple as raising the level of your yard (to match that of your neighbors') or changing the grade (or both).

Thanks succeed - I will take a look and try to get an idea as to how the water is meant to flow.
I'm also wondering about aerating and how much it will help...?

Sutherland05,
I think you might be getting ahead of yourself a bit. I hope you understand the importance of addressing your grade/slope/drainage problems first. Protecting your home from excess moisture is critical. And to do so, you may end up having to dig up or build up some portion(s) of your lawn anyway. So it is a little premature to talk about aeration.
First see what type of work it will require to fix your drainage issue, and in addressing that, you may find you have a chance to incorporate soil improvements as you go. For instance, if you end up finding that you need to add 6" in height to some areas of your backyard, and/or need to remove that amount from other areas of your backyard, then that in itself might give you the opportunity to get rid of some weeds, improve the soil composition, improve drainage, and maybe even improve the type/variety of grass in your backyard. Suddenly aeration becomes a minor consideration.
So before rushing to actively DO things, perhaps slow down and research what you really need, and make plans to address those issues one by one.
Note: Core aeration is not necessarily the right solution for everyone. There are some people who would tell you that there are better solutions for hard soil for less work/cost. There are some people who would tell you that aeration will just end up spreading the weed seed and that the effects of aeration will not be long lasting. And there will be other people who will tell you core aeration is the best thing they've ever done for their lawn. You can read more about the controversies of core aeration here on this forum.
But first, please address your drainage issues - fixing them first might change a lot of what is currently going on with your backyard.

So it sounds like I could try just adding some topsoil to some areas of the lawn where it is low and just try to divert the water to the driveway?
I definitely think we could use some height in the lawn, as there are lots of low spots - so would I just add topsoil here and there throughout the year? Wait and watch?
Thanks!

I would go down to the big box store, Lowes, Home Depot, etc. and get the longest 1X2 they have and a level. Sight down the board to make sure it is straight, Or get a laser level, but they become hard to read in sunlight.

Using the board and a level measure the drops in the length of the board to know where the high spots and the low spots are. I would start with the house and work out then from side to side. Mark the measurement points so you can work from them later. With a little impromptu surveying, you will have defined where your problems are and what you have to do to correct them.

Some time with slight elevation changes the perception of the problem is effected by the surroundings and is not the same as the actual problem.

As said above the moisture problem in the house is more important than having an instant perfect lawn.

SUTHERLAND05 YOU'RE NOT LISTENING!

If you have water draining into your house, you already have too much topsoil. Whatever you do, do not bring in any more until you have spoken to at least three professionals drainage experts. You're likely going to be hauling soil away in big dump trucks to get your drainage right. You need to call three professional landscapers who do "finish grading." Tell them you have water in the basement and ask them what do do. Let them talk and you quietly listen. PLEASE let them talk. You seem to have all the answers and are not listening to the advice you've gotten here. One of the three will be full of BS, hopefully one of the other two will convince you what you need to do.

Your drainage is your only problem at this point. You can lose your house and insurance may not pay for it if you don't get your drainage problem corrected. Don't even think about gardening until next fall. You're going to have a big mess until then. Besides with a big dog, you aren't going to have a lawn anyway. Think in terms of several inches of mulch.

Ok so I feel like a better explanation is in order for everyone to be on the same page here. I am on this forum to find out what I can do on my own, not with a professional grader and dump trucks, etc. Also, this is not my forever home, so I'm not interested in doing everything to the nth degree so I can lose money on a slightly less wet house. The area of the city I live in is known to be wet, and as I explained, the weeping tile, sump pump and dehumidifier are taking care of the house.
I am not ignoring the fact that I need to divert as much water as I can, but fixing my lawn is not going to be my life's mission for the next two years. What I want to do now is eliminate the soggier parts of the lawn, reduce the dry and cracked summer lawn and finally, make the lawn more grass than weeds. So while I gradually build up the lawn in certain areas, thereby diverting the water around the house, I am hoping that fertilizing, aerating and overseeding will be steps in the right direction.
Sorry for not being clear enough in the first post.