Sure, you can do that. I would mow your lawn, then spray everything you want dead with Round-Up. Then you can seed anytime after that, you only need to wait for the Round-up to dry.

In the fall repeat tiemco's advice. If you have a budget to consider, spend the most on the fall renovation. It will be the one that you'll see next spring. In other words, in the fall spend for better seed and take your time prepping if you can. And don't wait too late. Fall might come in August. Start when the summer heat breaks and the evenings cool off.

Thank you for the advice! I'm glad to hear I can do something now. What is a good source for better seed? I don't even know what kind we need. The grass seems to have died after we pruned our oak tree. It now gets more sun than it used to.
Oh, another question. Will Round-up kill the grass too? The clover is very interspersed with the grass.

Round-up kills pretty much everything that's green: weeds, grass, etc. Grass usually only dies from too much sun when it doesn't get enough water. Most of your better seed comes from independent seed sellers, not the big box stores. Many people source their seed via the internet. Choosing what type of grass to grow is up to you, and what you think would do best in your environment. Since this is a spring seeding, and you live in the city, tall fescue might be your best option. Kentucky bluegrass produces nicer turf in my opinion, but it is much better suited to a late summer seeding since it takes longer to germinate and mature. Perennial rye is the fastest to maturity, but it can be a bit thirstier.

Thanks again for all of your advice. Lack of water certainly could have been part of the problem, since we are not always so diligent about watering. I did search around for some independent seed sellers and found one that sells a sun & shade mix. It seems to contain a bit of each kind that you mentioned, so hopefully something would take.

Here is a link that might be useful: sun & shade mix

I am not a fan of "northern" or "tri-mixes", please excuse me if my previous post gave you that impression. I was only pointing out, very briefly, what your options are. If your yard has some full sun, and light to moderate shade (4-5 hours of sun) then tall fescue or a TF/KBG mix will perform well. Perennial rye prefers full sun, and will thin out in shade over time. Fine fescues are somewhat of a niche grass that are used in moderate to densely shaded lawns. In full sun and hot conditions in the summer they tend to go dormant. The mix you referenced will have three different types of grass, and can look inconsistent especially if you mow infrequently.

Okay, I will look for those options. Pardon my ignorance!

>tiny Brooklyn

With a tiny Brooklyn lawn, I'd go for it anytime. :-) It's far enough north that the heat won't do in seedlings. Once you do it, be a bit more faithful with the watering and mowing to keep up the good results.

It is unfortunate that Round-Up is getting their message across much better than Weed-B-Gone. The difference is that Round Up kills everything and WBG only kills broadleaf plants like clover. WBG leaves grassy plants alone. The product most people want for a lawn is WBG. It also seems like the WBG product most people want is Weed-B-Gone for Chickweed and Clover. Those words are in the name of the product, not buried in the fine print on the label.

I haven't had much luck with anything to kill clover, even Weed-B-Gon for Chickweed and Clover. If your lawn is tiny, you might consider pulling the clover. It crawls along just above or just below the soil line, and if you loosen the soil around it with a weeding tool (I like a swan-neck weeder) you can gently pull out long strands of it. My neighbors think I'm crazy, but it's quite addictive. :-)