Lysander is suspicious because Demetrius has been in love with Helena, but suddenly wants to marry Hermia.
Lysander fears Demetrius’s true intentions. He loves Hermia, but her father wants her to marry Demetrius. Lysander fears that Demetrius has only made a deal with her father, and does not really love Hermia.
Lysander does not immediately impeach Demetrius’s character before Theseus. At first, he tries to reason with Demetrius.
You have her father's love, Demetrius;(95)
Let me have Hermia's; do you marry him. (Act 1, Scene 1)
Demetrius does not seem to be a trustworthy sort, but he asks Lysander and Hermia to give each other up so he and Hermia can be together. Hermia will have none of it. Lysander gets frustrated with Demetrius. He reminds everyone that Demetrius was recently seen in the arms of another girl.
Demetrius, I'll avouch it to his head,
Made love to Nedar's daughter, Helena,
And won her soul; and she, sweet lady, dotes,(110)
Devoutly dotes, dotes in idolatry,
Upon this spotted and inconstant man. (Act 1, Scene 1)
Theseus remembers this, and decides to talk to Demetrius and Egeus privately. Lysander does not want Hermia to marry Demetrius, not only because he loves her but because he does not think Demetrius is faithful.
In the end, Lysander decides to run away with Hermia. He will go where her father and Theseus cannot reach them, and they will live happily ever after. This plan proves that Lysander is not “spotted and inconstant” and instead will risk his life to stick to Hermia’s side.