Great answer above with the Bancomer account.   I've got a good impression overall of Bancomer, though I don't care for our local branch in Oaxaca, friends elsewhere prefer them. Luckily our US bank (USAA) reimburses (some) foreign ATM charges, so we don't need to transfer. We just withdraw pesos using ATM from our US bank with good exchange rates. If we want money in the local bank we deposit the pesos.
Here on the coast of Oaxaca, our while family uses ScotiaBank, which has a good free "checking" account and a clunky but secure and usable online banking site.  We have found Scotia ATMs generally available. Transferring money between Scotia accounts is easy online and we do that a lot for the kids budgets when they are away at school.  Overall, Scotia suits us and are currently the ones who try harder in our area.
HSBC is an international chain like Scotia, bigger and more presence, but their Mexican version is not so great in many ways. In our area, at least, they have a bad attitude and are the most often rumoured to cheat or decieve their customers. On the other hand, they do offer a lot of special services, like cashier's checks in US dollars, if you ever needed that ! I can't recommend them.
Those are the only ones I can really comment on. Oh, and Banamex has a bad rep around here, but I've got no experience personally.Â
By the way, I've found that outside of the biggest cities, most Mexicans are unfamiliar with personal checks. They hardly used banks in that era and society was probably too sketchy for them to work anyway. People started using banks in numbers after ATMs came out. Same as phones. Very few Mexicans have ever had landlines, but almost all have cells now.
Another interesting note is that Mexico actually has a superior (to the US) interbank system for transfers and paying long distance. It's common and easy here and is usually free. You can also pay people by personally depositing in the bank's account wherever you instead of sending a check. It's commonly done.  When you buy on Mercado Libre (Mexican Ebay) the seller gives you his account number and bank. Either you send it to him from your bank online or you go to whatever their chain is - Bancomer, Banamex etc. and fill out a slip, handover the money and it appears anywhere in the country as now being in their account. It might even work at a different bank chain than theirs or yours, never tried.Â
Anyway, Bancomer is probably a good bet in general. I understand that they have good internet banking.
Cheers