Photo is Brookite, Moses Hill, Magnet Cove, Arkansas
Now that you have inspected the collection and made an assessment my preference is always to give a written offer with a "null and void" date usually 2 weeks hence. In the offer description you should include how you will pay. Even if you come to agreement immediately this written offer will stand as a bill of sale.
I recommend to collectors never to accept any kind of financing or paying forward arrangement, it is not in their best interest. If you need financing I would suggest seeking third party financing. (I usually give a 10% down cash (truly cash) deposit and the remaining amount in installments as I pack. So if it is going to take my 3 loads to haul away. I give 10% down then 30% by check with each load I take. That needs to be detailed in the offer. Of course if it is a one load collection payment is due the day of moving.)
If the offer is not accepted immediately be old school - send a hand written note of thanks the next day. The non-monetary part of selling a collection is very important. The collector spent a lot of time and effort into putting the collection together it is important to them to know that work is appreciated.
Be prepared for a bit of negotiation. You may be competing for the collection and may be in a better position to compete than a dealer. The dealer needs to resell and their offer will reflect the wholesale value. Refer to your notes when considering the counter offer.
So there it is. The hardest part about buying collections is finding one for sale. With a healthy checkbook and a gracious respect for other people it is really not that hard.