No, it won't work for the most part
When you cast Detect Poison and Disease you can sense poison and disease within range of yourself. Nothing in the spell description states that this is sight based:
For the duration, you can sense the presence and location of poisons, poisonous creatures, and diseases within 30 feet of you. You also identify the kind of poison, poisonous creature, or disease in each case.
Detect Magic is different though, it does work in a sight based manner:
For the duration, you sense the presence of magic within 30 feet of you. If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.
However, the rules for the spell Find Familiar state:
Additionally, as an action, you can see through your familiar's eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses.
Using the sight ability of Detect Magic takes an action. Seeing through your familiar's eyes also takes an action. You can't take two actions at the same time so you can't do this.
Casting either type of detect spell on your familiar isn't possible either as both have a range of "self" and you can only deliver touch spells through a familiar. You could have them use a magical item, if you have one, that has the effect of detect magic and you'd be able to see any magical items they can see.
As pointed out by V2Blast and Dale M, you can manage two actions in a single turn if you happen to have the action surge ability granted by the fighter class. It doesn't seem that would work though as Detect Magic states that you can only use the ability to see the magic if you can sense magic within 30 feet. But activating the ability to see and hear through your familiar makes you blind to your own senses which would include the magical sensing ability granted by the spell. Since you won't be able to meet the requirement to sense the magic, you won't be able to see the magic.