A limit game with a $1 maximum bet would be a .50-1.00 limit game. Blinds for this limit would be .25 & .50, the betting limit preflop and on the flop is .50 and the limit on the turn and river is 1.00.
First you figure out who's to deal and have them shuffle and then offer to the person on their right if they want to cut the deck. The first person to his left is the small blind(.25) and the second to the left is the big blind(.50) - these blinds are forced bets or "antes" to get the action started. After everyone has two cards the person to the immediate left of the big blind is first or "under the gun" and has to fold, call .50, or raise .50(which makes it a dollar total). Everyone after him does the same but can only raise by .50 over the highest bettor. I suggest a 4 raise maximum.
Next is burn one card and flop 3 cards. The first person to act is now the the first person to the left of the dealer(first person who's still has cards, not the person to the left of the big blind like in round 1). His/her options are to bet .50 or check. The action continues around the table clockwise with a .50 raise maximum. Next is the turn card(4th card) - burn one and turn it. Action again is to the first person to the left of the dealer, but now the limit is 1.00. All bets and raises on this round are in incerments of $1.00. The river(5th card) has the same $1.00 betting/raising limit.
After all the action is done on the river next is the showdown. Generally the person who was the last to bet or raise shows their cards first. Then all the other "callers" can turn theirs over to see who has the best hand. Once a hand is turned over "cards read". ie. If someone turns their hand over and says "two-pair" but it's really a full house... that's what they have. Both cards have to be shown on the table face-up and the best five card poker hand(made from their two cards and the 5 on the board) wins the pot. The person who was the small blind now becomes the next dealer and deals another hand!
All that said, we don't play a fixed limt, we play a spread limit. In your case how that would work is the blinds would be .25 and .50 but everyone could bet or raise anywhere between .25 and $1 on any round(except preflop where the action is already to .50)
Two more things: 1. I feel that allowing the checkraise is a mandatory part of poker - I don't care if your mother is sitting at the table. 2. I hope this helps, feel free to ask any more questions.
Now, shuffle-up and deal!!!