How do I fill out the hardship 401K withdrawal? Do I put the expenses as Distributions for eligible emergency expenses?

If you are asking "How do I get a hardship withdrawal from my employer?" then you have to ask the employer.  Every employer is allowed to decide what hardships they will allow.

If you are asking "How do I report a hardship withdrawal on my tax return?" you should have gotten a 1099-R from the plan.  Enter it in the program.  Click next to get to some followup questions about penalties.  Remember that all 401k withdrawals before age 59-1/2 are subject to regular income tax plus a 10% penalty.  There are some specific exceptions to the 10% penalty, but a "hardship" from your employer is not enough.  The penalty exceptions are very specific.  See this page,

https://www.irs.gov/retirement-plans/plan-participant-employee/retirement-topics-exceptions-to-tax-on-early-distributions

For example, you can claim an exemption for up to $1000 for a personal emergency.  If you withdrew more than $1000, you will still pay the penalty on the rest of the withdrawal.