Disclaimer: I am a beginner in both judo and physical culture. My views on strength, conditioning, and technique should be viewed with skepticism.
You're right that training once a week is insufficient. Most people won't see much progress in either physical condition or skills at less than two days a week minimum. I'll address solutions in the context of the "struggle points" that you mentioned.
Chin-ups and a Workout Routine for Judo
You are right to be concerned with strength, and chin-up strength in particular. While judo is a sublime skill and technical art, it relies on the body being strong and agile. Pulling strength is of particular use in judo. (The first movement of nearly all uchikomi practice is "pull your partner off balance", is it not?)
I recommend an all-around strength training program for judo, such as described in this answer. For further technical reading on this topic, see Practical Programming by Rippetoe and Kilgore. If you're training judo once a week and running twice a week, and assuming you're eating right and getting enough sleep, then two lifting sessions per week is probably about right.
You might not feel ready for that yet, or you might want to focus on the smaller goal of chin-ups, in which case my advice in this answer is relevant: train your strength as generally as possible, but pull-ups with a gi are king. (A towel will serve admirably in a pinch.) Since you can't yet do a pull-up, a goal of three in a row is a fair one.
- Start with a bar and work up to gi or towel pull-ups
- Remember to get a full range of motion: arms fully extended and relaxed at the bottom, chest touching the bar at the top.
- Start with negatives: jump up to the top and lower yourself down as slowly as possible. Keep trying to resist the downward movement all the way to the end.
- One method that I've seen espoused, but have not tried, is holding the top of the chin-up for a 10-count before lowering oneself down.
At-home technical training
My biggest concern for training for judo at home is not ingraining bad habits. To that end, most of my solo training is for strength, power, conditioning, mobility, or agility. That can include lifting weights (squats, deadlifts, pull-ups, power cleans, presses), sprints and unusual exercises to make myself tired very quickly, morning warm-ups and stretches, yoga, and "pre-hab" exercises. So I am not very well acquainted with at-home technical training, such as discussed in this question.
That said, I have tried the following, which may have been marginally productive:
I have heard that grappling dummies are fun, but I have not used one. My understanding is that they are better for drilling well-known technique, rather than figuring out new possibilities or trying to turn a technique one barely knows into a technique one is good with.
Running for Judo
Running twice a week is great, but consider switching to or adding sprints. Distance running is good stuff, but judo (and most athletics in general) benefit greatly from the power and metabolic training of short-duration, high-intensity bursts of effort. My method, adapted from my coach's advice, is a brief dynamic warm-up followed by a half-mile run. I then sprint two to six times on the way back, with the distance of the sprints inversely proportional to the number I intend to do: 50 yards each if I'm going to do 5 or 6; 100 yards each if I'm only going to do a couple. I let myself recover completely or almost completely between sprints. I'll throw in some lunges and karaoke runs and the short workout is over.