It doesn’t matter when you get paid. For anybody and everybody, no matter how often you get paid per month or when, you do the same thing each time you get paid. You ask yourself, “What does this paycheck need to fund until I get paid again?” You then fill in your categories based on this question until To be Budgeted is 0. (Rule #1)

Having said that, many of us like to fund a whole month at once while also getting approximately a month ahead of our paychecks. We call this breaking the paycheck-to-paycheck cycle. For you, that would mean budgeting all of next month with the paycheck you receive on the 6th of this month. To do this, you can either draw from existing savings or save up the funds needed to fund the current month over time. 

Goals don't work well if you're removing funds from the category in the current month. They're hardly necessary, though, and a number of people don't use them at all. For your example:

• In the January budget, there will be $50 available on Jan. 5. Move that to whatever high-priority category you like. (Physical movement of funds to a savings account is a separate consideration. *)
• On Jan. 6 when allocating your paycheck, add $300 to the Grocery budget entry. I would click in the budget cell, then type +300 and hit Enter to leverage the built-in calculator. Tip: put the desired funding level right in the category name for reference (Groceries $300/mo).

One option that would allow the use of Goals is to remove the funds from the category in last month's (Dec's) area, but if you spend on Jan. 1-5, that impacts the amount you should remove. At the least, I suggest you open a second browser window for reference to the current month so you can at least see the correct amount (balance as of Jan. 5) to remove. Again, use the arithmetic operator; e.g., -50 <return> to remove $50. 

This revisit of the previous month is typically necessary anyway as Import or Reconciliation will often pull a sneak attack and introduce overspending with transactions that post in last month's area.

Regardless of which of these you pursue, I find one very efficient way to "sweep" the leftovers is to remove all the balances first (putting those funds in To Be Budgeted), fix any overspending, then put the lump sum into the high-priority category.

As Superbone mentioned, getting to the point where you can align your budgeting with the calendar will make things a bit easier. Many will use part of their income replacement/emergency fund to do that. (If you did lose your job, that money would already be covering your categories in the current month, since your paycheck would have gone toward next month's categories.) Here are a couple workflows you might find useful. I suggest using the second as you work toward getting ahead (progress is obvious), and the first once you have the entire month covered on the 5th (before budgeting your check).

(*) There is no need to tie a category (or category group) to your savings account balance. Indeed, doing so will lose out on interest. Location of money is independent of purpose, and dollars are interchangeable. Just keep enough in Checking to cover upcoming expenses (running balance and scheduled transactions make this trivial to assess) and shuffle the rest to the higher-rate account.

Helter Skelter said:
If I have category (let's say called "savings) where I put all the money that I didn't spend then I still need to make the transaction of moving that amount from the checking to the savings account. If I am correct...

No, that is not correct. Read the article dakinemaui posted above about location versus purpose. The main purpose of a savings account is to increase the interest earned on the bulk of your budgeted funds that aren't required for near term spending. You just need enough in your checking account to pay your upcoming bills. The scheduler and running balance can help with illuminating that. You can then transfer extra funds to your on-budget savings account in one fell swoop, not tied to anything in particular and no category needed.

Helter Skelter said:
it is convenient for me to budget based on my payday

No, it's most convenient to decouple your payday from your budgeting and budget in one month chunks. Budget for all upcoming expenses and true expenses (Rule 2) and the rest can go into savings categories.

Helter Skelter said:
The solution that I have in my head is to enter the transactions of the first days of the next month (until I get paid) in the previous month

Fudging the dates? Seems easier to bring up a second window to facilitate sweeping in last month's area as previously described.

Helter Skelter said:
I am already not on the paycheck-to-paycheck cycle because I spend less money than I make and I have savings.

In that case, why not use some of those savings to advance where you budget your paycheck? Consider, you get paid on the 6th. Sometime between the 6th and the 30th, you hit the Underfunded button in next month's area and everything is filled out. Now THAT'S convenient.