Check first to make sure your bank will accept an ACH push instead of direct deposit. Try searching Google for "ACH push" and the bank's name to see if anyone else has posted that it will work, or ask your bank directly.
You need two bank accounts: your regular account, and the new account that needs an ACH push. Go online to your regular bank account and link the new bank account to it. Your regular bank will ask for the account number and the routing number of the new bank. Your bank should provide that information on its website, but it is readily available on your checks. The nine-digit number in the lower left-hand corner of your personal check is the bank's routing number. The 10-digit number to the right of the routing number is your account number.
Your regular bank's website will walk you through the linking process, but you will probably need to verify the link. Typically, your regular bank will automatically deposit two small amounts into your account at the new bank, and when they've cleared in two or three days, you must report the amounts to your regular bank. When your new bank account is successfully linked and verified with your old account, you're ready for the next step.
On the website of your regular bank, request a transfer of money to the new bank. That's an "ACH push." An amount as small as $5 or $10 may be enough to satisfy a direct deposit requirement, unless there's a larger minimum. You may be able to request that the transfer be done monthly automatically. Otherwise you'll need to remember to do it each month.