How should I handle putting my own money into the business account so I can avoid double taxation?
You are mixing two unrelated topics. Double taxation has nothing to do with your bank accounts.
Double taxation refers to the fact that income is taxed twice. This occurs when a corporation's profits are taxed at the corporate level, and then a second time when distributions are made to the shareholders/owners at the personal level.
To tie double taxation to an LLC it's worth explaining an LLC's tax structure choices. An LLC has the flexibility of choosing its tax structure. The two choices are corporate taxation or pass-through taxation.
Pass-Through Taxation: The business profits and losses are passed through to the owners, and taxed at the owners' personal income tax rates. The taxation will occur even if no profits are distributed to the owners (i.e. all profits are left in the business bank account). For a bit more information on how pass-through taxation works, see Simple Overview of LLC Pass Through Taxation.
Corporate Taxation: The business is treated as a separate entity. As a result, the business will pay income taxes on profits at the corporate tax rate, which may be lower than the owner’s individual tax rate. In addition to that, the owner will also have to pay income taxes on the portion of profits distributed to him, at his personal income tax rate. This is what leads to double taxation for the owner, because he is paying taxes twice - once as personal tax, and once as corporate tax.
So, if your LLC is setup as a pass-through entity (the default option) you don't have to worry about double taxation. However, if your LLC is taxed as a corporation, double taxation will be an issue.
If I understand you correctly, it seems like what you're really concerned with is how the money you invest into your LLC will be taxed. The answer is that you don't pay taxes on the money you put into the business. The reason is that it is assumed that you have already paid taxes on that money. The money you put into the business is considered your equity in the business, not income. You will, however, have to pay taxes on your profit.
To address your example:
I've received a check for Christmas from a relative for personal enjoyment. I wish to instead put it into my business so I can buy equipment. Should I deposit it directly into the business account, or deposit it into my personal account and transfer the money over? Does it even matter?
You should deposit it into your personal account and then transfer it to your business account. In fact, unless the check is made out to cash, you shouldn't be able to deposit a check made out to you into your business account.
Yes, it matters. You should be deliberate in your transfers. If you get in the habit of moving small amounts of money from one account to the other for no apparent reason you could "pierce the corporate veil". That means that the government could decide that you are not operating as a separate entity, and revoke your limited liability protection.