If your total income from all sources for the 2019 tax year is less than $12,300 you do not need to bother filing a tax return.

However, if you have self-employment income of more than $400, then you are required to file a tax return. With only alimony income, you would not be required to file a return at all. Though I do question how you survived the entire tax year on only $5,250.

To register for the stimulus without filing at tax return, go to https://www.irs.gov/coronavirus/non-filers-enter-payment-info-here

You are not required to file a tax return if your only income is the $5,250 of alimony that you received. 

You can provide the necessary information to receive an economic impact payment ("stimulus check") here: Non-Filers payment registration.

Addressing only the alimony question

The Tax Cuts and Jobs Act enacted new tax rules regarding spousal support payments, also known as alimony. In divorces finalized after January 1, 2019, the person paying spousal support can no longer deduct the amount from their taxes. For recipients, spousal support payments are no longer considered taxable income.Jan 15,2019

https://www.irs.gov/forms-pubs/clarification-changes-to-deduction-for-certain-alimony-payments-effec...

For recipients, spousal support payments are no longer considered taxable income.

If that's the case, that means you have less than $4,150 of taxable income and someone else "could" qualify to claim you as a dependent on their tax return. If someone else does "in fact" claim you as their dependent, then you do not qualify for a stimulus payment.